Multipliers and equivalence of functions, spaces, and operators

Abstract

By interpreting the well-known Brown–Halmos theorem for Toeplitz operators in terms of multipliers, we formulate a Brown–Halmos analogue for the product of generalized Toeplitz operators, defined as compressions of multiplication operators to closed subspaces of $L^2({\mathbb {T}})$. We use this to define equivalences between two operators in that class by means of multipliers between the spaces where they act. Necessary and sufficient conditions for such an equivalence to be unitary or a similarity are established. The results are applied to Toeplitz and Hankel operators, truncated Toeplitz operators, and dual truncated Toeplitz operators.

1 Introduction

A well-known theorem of Brown and Halmos [5, Thm. 8] says that if $T_{\varphi }$ and $T_{\psi }$ are two Toeplitz operators on the Hardy space $H^2$ (definitions below), then $T_{\varphi } T_{\psi } = T_{\varphi \psi }$ if and only if either $\varphi \in \overline{H^{\infty }}$ or $\psi \in H^{\infty }$. If one makes the observation that $\varphi \in \overline{H^{\infty }}$ if and only if $\varphi (H^2)^{\perp } \subseteq (H^2)^{\perp }$ and that $\psi \in H^{\infty }$ if and only if $\psi H^2 \subseteq H^2$, one sees a path towards generalizing this result to a broader class of operators by using the notion of multipliers between subspaces of $L^2$.

To describe this class of operators, let $L^{2}$ denote the standard Lebesgue space of the circle and $L^{\infty }$ denote the essentially bounded functions in $L^2$. For $\varphi \in L^{\infty }$, let $M_{\varphi }: L^2 \rightarrow L^2$ denote the multiplication operator $M_{\varphi } f = \varphi f$. For two closed subspaces ${\mathcal {H}}_1$ and ${\mathcal {H}}_{2}$ of $L^2$, the generalized Toeplitz operator from ${\mathcal {H}}_1$ to ${\mathcal {H}}_2$ with symbol $\varphi \in L^{\infty }$, is defined by

$$\begin{aligned} T_\varphi ^{{\mathcal {H}}_1,{\mathcal {H}}_2}: {\mathcal {H}}_1 \rightarrow {\mathcal {H}}_2, \quad T_\varphi ^{{\mathcal {H}}_1,{\mathcal {H}}_2} = P_{{\mathcal {H}}_2} M_\varphi |_{{\mathcal {H}}_1}, \end{aligned}$$

(1.1)

where $P_{{\mathcal {H}}}$ denotes the orthogonal projection from $L^2$ onto the closed subspace ${\mathcal {H}}$. We will use the notation ${\mathscr {T}}({\mathcal {H}}_{1}, {\mathcal {H}}_2):= \{T_\varphi ^{{\mathcal {H}}_1,{\mathcal {H}}_2}: \varphi \in L^{\infty }\}$ and ${\mathscr {T}}({\mathcal {H}}):= {\mathscr {T}}({\mathcal {H}}, {\mathcal {H}}).$

These generalized Toeplitz operators are particular cases of general Wiener–Hopf operators [15, 41]. When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = H^{2}$, the standard Hardy space of the disk [21, 34], the set ${\mathscr {T}}(H^{2})$ is the set of Toeplitz operators [4]. When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = (H^2)^{\perp } = L^2 \ominus H^2$, ${\mathscr {T}}((H^{2})^{\perp })$ is the set of dual Toeplitz operators [42]. When ${\mathcal {H}}_1 = H^{2}$ and ${\mathcal {H}}_2 = (H^{2})^{\perp }$, the set ${\mathscr {T}}(H^{2}, (H^{2})^{\perp })$ becomes the set of Hankel operators [38, 39]. For inner functions $\Theta _1, \Theta _2$ and their corresponding model spaces ${\mathcal {K}}_{\Theta _j} = H^{2} \ominus \Theta _j H^{2}$, $j = 1, 2$, the operators in ${\mathscr {T}}({\mathcal {K}}_{\Theta _1}, {\mathcal {K}}_{\Theta _2})$ are the asymmetric truncated Toeplitz operators [6, 11, 40]. When ${\mathcal {H}}_j = ({\mathcal {K}}_{\Theta _j})^{\perp } = L^2 \ominus {\mathcal {K}}_{\Theta _j} = \Theta _j H^2 \oplus (H^2)^{\perp }$, $j = 1, 2$, then ${\mathscr {T}}(({\mathcal {K}}_{\Theta _1})^{\perp }, ({\mathcal {K}}_{\Theta _2})^{\perp })$ is the set of asymmetric dual truncated Toeplitz operators [7, 8, 16,17,18]. We will discuss the above classes of operators throughout this paper as motivating examples of generalized Toeplitz operators.

Our first result is Theorem 2.7 where we prove the following generalization of the Brown–Halmos theorem: if ${\mathcal {H}}, {\mathcal {K}}_1, {\mathcal {K}}_2$ are closed subspaces of $L^2$ and $\psi , \varphi \in L^{\infty }$ and either $\psi {\mathcal {H}}^{\perp } \subseteq {\mathcal {K}}_{2}^{\perp }$ or $\varphi {\mathcal {K}}_1 \subseteq {\mathcal {H}}$, then

$$\begin{aligned} T_{\psi \varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = T_{\psi }^{{\mathcal {H}}, {\mathcal {K}}_2} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {H}}}. \end{aligned}$$

In this generality, the converse is not longer true. However, we do have a partial converse given in Proposition 3.15.

Since the domains and codomains of $T_{\varphi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}$ and $T_{\psi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}$ are different, it is impossible, in general, to explore the familiar concepts of similarity and unitary equivalence of operators. Thankfully, there is a useful substitution. Indeed, $A \in {\mathscr {B}}({\mathcal {H}}_1, {\mathcal {H}}_2)$ and $B \in {\mathscr {B}}({\mathcal {K}}_1, {\mathcal {K}}_2)$ are equivalent if there are bounded invertible operators $E: {\mathcal {K}}_{2} \rightarrow {\mathcal {H}}_{2}$ and $F: {\mathcal {H}}_{1} \rightarrow {\mathcal {K}}_{1}$ such that

$$\begin{aligned} A = E B F. \end{aligned}$$

(1.2)

The above notion of equivalence has been used to study spectral properties of operators and the solvability of singular integral equations [2, 12, 30, 31, 37, 41]. For instance, the main motivation behind Wiener–Hopf factorization, and its varied applications, relies on the fact that it provides a useful equivalence between a Toeplitz operator (on the Hardy space) with a nonvanishing continuous symbol and a Toeplitz operator with a more easily understood monomial symbol. One can show that equivalent operators are simultaneously Fredholm (with the same Fredholm defect numbers) and the range, kernel, and inverse of one operator is uniquely determined by the other. For example, a straightforward argument using the singular value decomposition will show that two $n \times n$ matrices, thought of as linear transformations on ${\mathbb {C}}^{n}$, are equivalent if and only if they have the same rank (see Proposition 6.8).

To discuss equivalence of generalized Toeplitz operators, let ${\mathscr {G}}\!L^{\infty }$ denote the set of invertible elements of the algebra $L^{\infty }$ (i.e., $a, 1/a \in L^{\infty }$). Then, if $a_1, a_2\in {\mathscr {G}}\!L^\infty$ satisfy ${\mathcal {H}}_1 = a_1 {\mathcal {K}}_1$ and ${\mathcal {H}}_2 = a_2 {\mathcal {K}}_2,$ Proposition 2.8 shows that the operators $E: {\mathcal {K}}_2 \rightarrow {\mathcal {H}}_2$ and $F: {\mathcal {H}}_1 \rightarrow {\mathcal {K}}_1$ defined by

$$\begin{aligned} E = T_{\overline{a_2}^{-1}}^{{\mathcal {K}}_2, {\mathcal {H}}_2} \quad \text{ and } \quad F = T_{a_1^{-1}}^{{\mathcal {H}}_1, {\mathcal {K}}_1} \end{aligned}$$

are invertible. Finally, if the functions $\varphi , \psi \in L^{\infty }$ are related by the identity

$$\begin{aligned} \psi = \overline{a_2} \varphi \, a_1, \end{aligned}$$

(1.3)

where $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$ and $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$, we will show in Theorem 2.12 that

$$\begin{aligned} T_\varphi ^{{\mathcal {H}}_1, {\mathcal {H}}_2} = E T_\psi ^{{\mathcal {K}}_1, {\mathcal {K}}_2} F, \end{aligned}$$

(1.4)

and thus the generalized Toeplitz operators $T_\varphi ^{{\mathcal {H}}_1, {\mathcal {H}}_2}$ and $T_\psi ^{{\mathcal {K}}_1, {\mathcal {K}}_2}$ are equivalent in the sense of (1.2). Moreover, they are equivalent by means of the operators E and F which, in themselves, are generalized Toeplitz operators. Both this theorem and Theorem 2.7, which are central to this paper, have simple algebraic proofs—but interesting and impactful applications.

When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {H}}$, ${\mathcal {K}}_1 = {\mathcal {K}}_2 = {\mathcal {K}}$, and $a_1 = a_2 = a$, the identity from (1.4) becomes

$$\begin{aligned} T_{\varphi }^{{\mathcal {H}}} = F^{*} T_{\psi }^{{\mathcal {K}}} F \end{aligned}$$

and it is natural to ask whether $F^{*} = F^{-1}$, making $F = T^{{\mathcal {H}}, {\mathcal {K}}}_{a^{-1}}$ a unitary operator, and hence making the generalized Toeplitz operators $T_{\varphi }^{{\mathcal {H}}}$ and $T_{\psi }^{{\mathcal {K}}}$ unitarily equivalent. We formulate conditions for when this occurs in Proposition 2.18 and Corollary 2.20, by reducing this question to that of characterizing the generalized Toeplitz operators from ${\mathcal {H}}$ to ${\mathcal {K}}$ which are equal to the zero operator.

In this paper, we focus on how the relations between generalized Toeplitz operators, induced by multipliers, are reflected in their kernels, their ranges, and their relationships to conjugations.

In Sect. 2 we outline our basic definitions of multipliers between subspaces of $L^2$ and set up the material needed to state and prove the first two main theorems of this paper (Theorems 2.7 and 2.12). Though these theorems have relatively simple algebraic proofs, their reformulations in terms of Toeplitz, dual Toeplitz, Hankel, truncated Toeplitz operators are quite interesting.

In Sect. 3 we focus on the relationships between the kernels and ranges of generalized Toeplitz operators acting between spaces that are related by multipliers, thus extending some known properties of classical Toeplitz and Hankel operators.

Section 4 addresses the question of whether the equivalence of operators obtained in Theorem 2.12 preserves complex selfadjointness of those operators, as it happens, for instance with invertibility and Fredholmness. We establish conditions on the operators E and F from Proposition 2.8 to induce “equivalent” antiunitary operators $C_{{\mathcal {H}}}$ and $C_{{\mathcal {K}}}$ on ${\mathcal {H}}$ and ${\mathcal {K}}$ respectively, and for selfadjointness with respect to $C_{{\mathcal {H}}}$ and $C_{{\mathcal {K}}}$ to be preserved under the operator equivalence of Theorem 2.12.

In the last three sections of the paper, we apply our results to various types of generalized Toeplitz operators.

In Sect. 5 we show that when ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {K}}_2 = {\mathcal {K}}_2 = H^2$ (the Hardy space), an equivalence of symbols defined by (1.3) can be used to establish isomorphisms between Toeplitz kernels and, in particular, between a Toeplitz kernel and a model space. We also give necessary and sufficient conditions for this isomorphism to be isometric. We will see plenty of examples of equivalent functions amongst the inner functions and unimodular functions. For example, any Frostman shift of an inner function $\Theta$ is equivalent to $\Theta$, and any two finite Blaschke products of the same degree are equivalent. It is also the case that any two Hölder continuous functions (with exponent in (0, 1)) with the same winding number about the origin are equivalent.

In Sect. 6 we first apply the previous results to reduce the study of a truncated Toeplitz operator on any finite dimensional model space to another operator of the same type on the simplest of model spaces $(z^n H^2)^{\perp }$, the polynomials of degree at most n. Secondly, we show that given any rational symbol in $L^{\infty }$, any two truncated Toeplitz operators with that symbol, defined on model spaces determined by equivalent inner functions, have the same essential spectrum. This generalizes a result of Crofoot [14, Thm. 14].

Finally, in Sect. 7, we use our results on generalized Toeplitz operators to completely describe the kernel of a dual truncated Toeplitz operator acting on the space ${\mathcal {K}}_{\Theta }^{\perp } = (H^2)^{\perp } \oplus \Theta H^2$, where $\Theta$ is a finite Blaschke product, whose symbol is not invertible in $L^{\infty }$. In particular, we show, somewhat surprisingly, that its kernel has no elements belonging to $\Theta H^2$ and in fact, this kernel is contained in ${\mathcal {K}}_{\Theta }^{\perp } \ominus \Theta H^2$.

2 Multipliers, projections, and equivalent operators

Let $L^2:= L^2(m)$, where $m = d \theta /2\pi$ represents the standard normalized Lebesgue measure on ${\mathbb {T}}$, and $L^{\infty } = L^{\infty }(m)$ denotes the set of essentially bounded functions on the unit circle ${\mathbb {T}}$. For two closed subspaces ${\mathcal {H}}$ and ${\mathcal {K}}$ of $L^2$, the function $a \in L^{\infty }$ is a multiplier from ${\mathcal {K}}$ to ${\mathcal {H}}$ if $a {\mathcal {K}}\subseteq {\mathcal {H}}$. In this section we will focus our attention on the case when $a \in \mathscr {G}\!L^{\infty }$, where

$$\begin{aligned} {\mathscr {G}}\!L^{\infty }:= \left\{ g \in L^{\infty }: \frac{1}{g} \in L^{\infty }\right\} \end{aligned}$$

(the invertible elements of the algebra $L^{\infty }$) and $a {\mathcal {K}}= {\mathcal {H}}$. In this case, the multiplication operator $M_{a}: {\mathcal {K}}\rightarrow {\mathcal {H}}$, defined by $M_{a} f = a f$, is bounded and invertible.

An important class of multipliers arises when ${\mathcal {H}}$ and ${\mathcal {K}}$ are both equal to $H^2$, the classical Hardy space

$$\begin{aligned} H^2 = \{f \in L^2: {\widehat{f}}(n) = 0 \text{ for } \text{ all } n < 0\}. \end{aligned}$$

In the above, $({\widehat{f}}(n))_{n = -\infty }^{\infty }$ is the sequence of Fourier coefficients associated with an $f \in L^2$. By means of radial boundary values on ${\mathbb {T}}$, we can also regard $H^2$ as the space of analytic functions on the open unit disk ${\mathbb {D}}$ whose Taylor coefficients belong to the sequence space $\ell ^2$. Standard facts about $H^2$ (inner-outer factorization [21, Ch. 2]) say that $\varphi H^2 \subseteq H^2$ if and only if $\varphi \in H^{\infty }:= H^2 \cap L^{\infty }$ and $\varphi H^{2} = H^2$ if and only if $\varphi \in {\mathscr {G}}\!H^{\infty }$, where

$$\begin{aligned} {\mathscr {G}}\!H^{\infty }:= \left\{ g \in H^{\infty }: \frac{1}{g} \in H^{\infty }\right\} \end{aligned}$$

(the invertible elements of the algebra $H^{\infty }$).

Another important example of multipliers appears when ${\mathcal {H}}$ and ${\mathcal {K}}$ are model spaces. A model space corresponding to an inner function I ($I \in H^{\infty }$ and $|I| = 1$ almost everywhere on ${\mathbb {T}}$) is

$$\begin{aligned} {\mathcal {K}}_{I}:= H^2 \cap (I H^2)^{\perp }. \end{aligned}$$

See [24] for the basics of model spaces. Multipliers between model spaces were initially explored by Crofoot [14] and then later in [22]. If $a \in {\mathscr {G}}\!L^{\infty }$ and $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$ ($\alpha$ and $\theta$ are inner), results from [14] show that $a \in {\mathscr {G}}\!H^{\infty }$. In fact, there is a complete description of these types of multipliers [14, Cor. 18].

Proposition 2.1

(Crofoot) For $a \in {\mathscr {G}}\!L^{\infty }$ and inner functions $\theta$ and $\alpha$, the following are equivalent:

• (a) $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$;

• (b) $a \in {\mathscr {G}}\!H^{\infty }$ and $\alpha {\overline{a}}/(\theta a)$ is a unimodular constant function on ${\mathbb {T}}$.

There can be unbounded multipliers from one model space onto another [22, Thm. 6.9] but in this paper we will only consider the bounded ones. The multipliers from one model space to itself, are just the non-zero constant functions [14, Prop. 12]. Finally, there may not be any multiplier between two given model spaces (e.g., when $\theta$ and $\alpha$ are finite Blaschke products of different degrees as this will correspond to finite dimensional model spaces of different dimensions [24, p. 116]). However, when $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$, the multiplier a is unique up to a nonzero constant factor [14, Cor. 13].

Model spaces ${\mathcal {K}}_{\theta }$ are kernels of Toeplitz operators, i.e., ${\mathcal {K}}_{\theta } = {\text {ker}}T_{{\overline{\theta }}}$ [24, p. 108]. For a $\varphi \in L^{\infty }$, recall the Toeplitz operator $T_{\varphi }$ with symbol $\varphi$ is the bounded operator on $H^2$ defined by

$$\begin{aligned} T_{\varphi } f = P_{+}(\varphi f), \end{aligned}$$

(2.2)

where $P_{+}$ denotes the Riesz projection [4, 20]. Recall that $P_{+}$ is the orthogonal projection of $L^2$ onto $H^2$ and is given via Fourier series as

$$\begin{aligned} P_{+}\left( \sum _{n = -\infty }^{\infty } {\widehat{f}}(n) \xi ^n\right) = \sum _{n = 0}^{\infty } {\widehat{f}}(n) \xi ^n, \quad f \in L^2. \end{aligned}$$

This allows us to formulate an equivalent version of Proposition 2.1 which will be useful in a moment.

Proposition 2.3

For $a \in {\mathscr {G}}\!L^{\infty }$ and inner functions $\theta$ and $\alpha$, the following are equivalent:

• (a) $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$;

• (b) $a \in {\mathscr {G}}\!H^{\infty }$ and $a = {\overline{\theta }} \alpha {\overline{h}}$ for some $h \in {\mathscr {G}}\!H^{\infty }$.

Proof

If $a \in {\mathscr {G}}\!L^{\infty }$ and $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$, then $a \in {\mathscr {G}}\!H^{\infty }$. Using [12, Prop. 2.16],

$$\begin{aligned} {\mathcal {K}}_{\alpha } = a {\mathcal {K}}_{\theta }&\iff {\text {ker}}T_{{\overline{\alpha }}} = a {\text {ker}}T_{{\overline{\theta }}}\\&\iff {\text {ker}}T_{{\overline{\alpha }}} = {\text {ker}}T_{a^{-1} {\overline{\theta }}}\\&\iff {\overline{\alpha }} = a^{-1} {\overline{\theta }} {\overline{h}} \quad \hbox { for some}\ h \in {\mathscr {G}}\!H^{\infty }. \end{aligned}$$

Thus, $a = {\overline{\theta }} \alpha {\overline{h}}$ with $h \in {\mathscr {G}}\!H^{\infty }$. Conversely, if $a \in {\mathscr {G}}\!H^{\infty }$ and $a = {\overline{\theta }} \alpha {\overline{h}}$ with $h \in {\mathscr {G}}\!H^{\infty }$, then ${\text {ker}}T_{{\overline{\alpha }}} = {\text {ker}}T_{{\overline{\theta }} a^{-1} {\overline{h}}} = a {\text {ker}}T_{{\overline{\theta }}}.$ Hence, $a {\mathcal {K}}_{\theta } = {\mathcal {K}}_{\alpha }$. $\square$

Returning to our general discussion of multipliers from ${\mathcal {H}}$ onto ${\mathcal {K}}$, we have the following useful results for the multipliers between the annihilators ${\mathcal {H}}^{\perp }$ and ${\mathcal {K}}^{\perp }$.

Proposition 2.4

• (a) For $a \in L^{\infty }$, the following are equivalent:

  • (i) $a {\mathcal {K}}\subseteq {\mathcal {H}}$;

  • (ii) ${\overline{a}} {\mathcal {H}}^{\perp } \subseteq {\mathcal {K}}^{\perp }$.

• (b) For $a \in {\mathscr {G}}\!L^{\infty }$, the following are equivalent:

  • (i) $a {\mathcal {K}}= {\mathcal {H}}$;

  • (ii) ${\overline{a}} {\mathcal {H}}^{\perp } = {\mathcal {K}}^{\perp }$.

Proof

Suppose $a {\mathcal {K}}\subseteq {\mathcal {H}}$. If $k \in {\mathcal {K}}$ and ${\widetilde{h}} \in {\mathcal {H}}^{\perp }$, then $\langle {\overline{a}} {\widetilde{h}}, k\rangle = \langle {\widetilde{h}}, a k\rangle = 0$ and so ${\overline{a}} {\mathcal {H}}^{\perp } \subseteq {\mathcal {K}}^{\perp }$. The argument above can be reversed. $\square$

Propositions 2.3 and 2.4 yield the following.

Proposition 2.5

For $b \in {\mathscr {G}}\!L^{\infty }$ and inner functions $\alpha , \theta$, the following are equivalent:

• (a) $b {\mathcal {K}}_{\alpha }^{\perp } = {\mathcal {K}}_{\theta }^{\perp }$;

• (b) $b \in {\mathscr {G}}\!\overline{H^{\infty }}$ and $\alpha b/(\theta {\overline{b}})$ is a unimodular constant function on ${\mathbb {T}}$;

• (c) $b \in {\mathscr {G}}\!\overline{H^{\infty }}$ and $b = \theta {\overline{\alpha }} h$ for some $h \in {\mathscr {G}}\!H^{\infty }$.

In what follows, ${\mathcal {H}}_1, {\mathcal {H}}_2, {\mathcal {K}}_1, {\mathcal {K}}_2$ will be closed subspaces of $L^2$ and, for a closed subspace ${\mathcal {H}}$ of $L^2$, $P_{{\mathcal {H}}}$ will denote the orthogonal projection of $L^2$ onto ${\mathcal {H}}$ (and of course $P_{{\mathcal {H}}^{\perp }} = I_{{\text {d}}} - P_{{\mathcal {H}}}$).

A theorem of Brown and Halmos [5, Thm. 8] says that for the classical Toeplitz operators on the Hardy space $H^2$ we have, for $\varphi , \psi \in L^{\infty }$,

$$\begin{aligned} T_{\varphi } T_{\psi } = T_{\varphi \psi } \iff \varphi \in \overline{H^{\infty }} \text{ or } \psi \in H^{\infty }\text{. } \end{aligned}$$

(2.6)

Since the condition $\varphi \in \overline{H^{\infty }}$ is equivalent to $\varphi (H^2)^{\perp } \subseteq (H^2)^{\perp }$ and the condition $\psi \in H^{\infty }$ is equivalent to $\psi H^2 \subseteq H^2$, the formula in (2.6) can be interpreted in terms of multipliers as saying that $T_{\varphi } T_{\psi } = T_{\varphi \psi }$ if and only if either $\varphi$ is a multiplier from $(H^2)^{\perp }$ into $(H^2)^{\perp }$ or $\psi$ is a multiplier from $H^2$ into $H^2$.

For generalized Toeplitz operators the “only if” part of the theorem cannot always be generalized since the symbols $\varphi$ and $\psi$ are not unique (see (2.26) and (2.27) below). Thus, the following theorem appears to be the natural generalization of the Brown–Halmos theorem for generalized Toeplitz operators. We will discuss this further in the next section.

Theorem 2.7

Let $\psi , \varphi \in L^{\infty }$. If either $\psi {\mathcal {H}}^{\perp } \subseteq {\mathcal {K}}_{2}^{\perp }$ or $\varphi {\mathcal {K}}_1 \subseteq {\mathcal {H}}$, then

$$\begin{aligned} T_{\psi \varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = T_{\psi }^{{\mathcal {H}}, {\mathcal {K}}_2} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {H}}}. \end{aligned}$$

Proof

If $\psi {\mathcal {H}}^{\perp } \subseteq {\mathcal {K}}_{2}^{\perp }$ we have

$$\begin{aligned} T_{\psi \varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}&= P_{{\mathcal {K}}_2} (\psi \varphi ) P_{{\mathcal {K}}_1}|_{{\mathcal {K}}_1}\\&= P_{{\mathcal {K}}_2} \psi (P_{{\mathcal {H}}} + P_{{\mathcal {H}}^{\perp }}) \varphi P_{{\mathcal {K}}_{1}}|_{{\mathcal {K}}_1}\\&= P_{{\mathcal {K}}_2} \psi P_{{\mathcal {H}}} \varphi P_{{\mathcal {K}}_1}|_{{\mathcal {K}}_1}\\&= T_{\psi }^{{\mathcal {H}}, {\mathcal {K}}_2} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {H}}}. \end{aligned}$$

If $\varphi {\mathcal {K}}_1 \subseteq {\mathcal {H}}$, then

$$\begin{aligned} T_{\psi \varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = P_{{\mathcal {K}}_2} \psi \varphi P_{{\mathcal {K}}_1}|_{{\mathcal {K}}_1} = P_{{\mathcal {K}}_2} \psi P_{{\mathcal {H}}} \varphi P_{{\mathcal {K}}_1}|_{{\mathcal {K}}_1} = T_{\psi }^{{\mathcal {H}}, {\mathcal {K}}_2} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {H}}}. \end{aligned}$$

$\square$

As we approach the second main theorem of this section, we note that it follows from Theorem 2.7 that each $a \in {\mathscr {G}}\!L^{\infty }$ for which $a {\mathcal {K}}= {\mathcal {H}}$ determines two closely related bounded invertible operators from ${\mathcal {K}}$ onto ${\mathcal {H}}$.

Proposition 2.8

Let $a \in {\mathscr {G}}\!L^{\infty }$ with $a {\mathcal {K}}= {\mathcal {H}}$. Then the operators

$$\begin{aligned} T_{a}^{{\mathcal {K}}, {\mathcal {H}}} = M_{a}|_{{\mathcal {K}}} \quad \text{ and } \quad T_{{\overline{a}}^{-1}}^{{\mathcal {K}}, {\mathcal {H}}} = P_{{\mathcal {H}}} {\overline{a}}^{-1} P_{{\mathcal {K}}}|_{{\mathcal {K}}} \end{aligned}$$

are invertible with

$$\begin{aligned} (T_{a}^{{\mathcal {K}}, {\mathcal {H}}})^{-1} = T_{a^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} \quad \text{ and } \quad (T_{{\overline{a}}^{-1}}^{{\mathcal {K}}, {\mathcal {H}}})^{-1} = T_{{\overline{a}}}^{{\mathcal {H}}, {\mathcal {K}}} \end{aligned}$$

(2.9)

which are generalized Toeplitz operators. Moreover,

$$\begin{aligned} T_{{\overline{a}}^{-1}}^{{\mathcal {K}}, {\mathcal {H}}} = [(T_{a}^{{\mathcal {K}}, {\mathcal {H}}})^{*}]^{-1}. \end{aligned}$$

Proof

The invertibility and the two equalities in (2.9) follow from Theorem 2.7 and the identities

$$\begin{aligned} T_{{\overline{a}}^{-1}}^{{\mathcal {K}}, {\mathcal {H}}}&= (T_{a^{-1}}^{{\mathcal {H}}, {\mathcal {K}}})^{*} = [(T_{a}^{{\mathcal {K}}, {\mathcal {H}}})^{-1}]^{*} = [(T_{a}^{{\mathcal {K}}, {\mathcal {H}}})^{*}]^{-1}. \end{aligned}$$

$\square$

The next two results relate the orthogonal projections $P_{{\mathcal {H}}}$ and $P_{{\mathcal {K}}}$ when ${\mathcal {H}}$ and ${\mathcal {K}}$ are related by multipliers.

Proposition 2.10

If $a \in {\mathscr {G}}\!L^{\infty }$ with ${\mathcal {H}}\subseteq a {\mathcal {K}}$, then

$$\begin{aligned} P_{{\mathcal {H}}} f = a P_{{\mathcal {K}}} a^{-1} P_{{\mathcal {H}}} f = P_{{\mathcal {H}}} {\overline{a}}^{-1} P_{{\mathcal {K}}} {\overline{a}}f \quad \hbox { for all}\ f \in L^2. \end{aligned}$$

Proof

Since $a^{-1} {\mathcal {H}}\subseteq {\mathcal {K}}$ we have that

$$\begin{aligned} P_{{\mathcal {H}}} f = a a^{-1} P_{{\mathcal {H}}} f = a P_{{\mathcal {K}}} a^{-1} P_{{\mathcal {H}}} f \quad \hbox { for all}\ f \in L^2. \end{aligned}$$

On the other hand, by Proposition 2.4, $a^{-1} {\mathcal {H}}\subseteq {\mathcal {K}}\implies {\overline{a}}^{-1}{\mathcal {K}}^{\perp } \subseteq {\mathcal {H}}^{\perp }$, and so

$$\begin{aligned} P_{{\mathcal {H}}} f&= P_{{\mathcal {H}}} {\overline{a}}^{-1} {\overline{a}} P_{{\mathcal {H}}} f = P_{{\mathcal {H}}} {\overline{a}}^{-1} P_{{\mathcal {K}}} {\overline{a}} P_{{\mathcal {H}}} f \quad \text{ for } \text{ every }\ f \in L^2. \end{aligned}$$

$\square$

Corollary 2.11

If $a \in {\mathscr {G}}\!L^{\infty }$ with $a {\mathcal {K}}= {\mathcal {H}}$, then

$$\begin{aligned} P_{{\mathcal {H}}} = {\widetilde{E}} (P_{{\mathcal {K}}} {\overline{a}} I_{{\text {d}}}) = (a P_{{\mathcal {K}}}) {\widetilde{F}}, \end{aligned}$$

where ${\widetilde{E}} = P_{{\mathcal {H}}} {\overline{a}}^{-1} P_{{\mathcal {K}}}$ and ${\widetilde{F}} = P_{{\mathcal {K}}} a^{-1} P_{{\mathcal {H}}}$ are such that $T_{{\overline{a}}^{-1}}^{{\mathcal {K}}, {\mathcal {H}}} = {\widetilde{E}}|_{{\mathcal {K}}}$ and $T_{a^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} = {\widetilde{F}}|_{{\mathcal {H}}}.$

We now formulate our main theorem regarding equivalence of generalized Toeplitz operators. Various reformulations of this will be provided below (for Toeplitz, Hankel, and truncated Toeplitz operators). In §6 we will see an application of this to the inverse of a truncated Toeplitz operator, and in §7 we reformulate this for dual truncated Toeplitz operators.

Theorem 2.12

Let ${\mathcal {H}}_1, {\mathcal {H}}_2, {\mathcal {K}}_1, {\mathcal {K}}_2$ be closed subspaces of $L^2$ and $a_1, a_2 \in {\mathscr {G}}\!L^{\infty }$ with $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$ and $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$. If $\varphi , \psi \in L^{\infty }$ with $\varphi = {\overline{a}}_2 \psi a_1$, then

$$\begin{aligned} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = E T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} F, \end{aligned}$$

(2.13)

where

$$\begin{aligned} E = T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} \quad \text{ and } \quad F = T_{a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_1} = M_{a_1}|_{{\mathcal {K}}_1}. \end{aligned}$$

Thus, the generalized Toeplitz operators $T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}$ and $T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}$ are equivalent.

Proof

The equivalence in (2.13) follows from Corollary 2.11 since we have

$$\begin{aligned} P_{{\mathcal {K}}_2}&= T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} P_{{\mathcal {H}}_2} {\overline{a}}_{2}^{-1} I_{L_2} \quad \text{ and } \quad P_{{\mathcal {K}}_1} = a_{1}^{-1} P_{{\mathcal {H}}_1} a_1 P_{{\mathcal {K}}_1} = a_{1}^{-1} T_{a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_1}. \end{aligned}$$

$\square$

Remark 2.14

By Theorem 2.7, the identity

$$\begin{aligned} T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = E T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} F, \end{aligned}$$

(2.15)

where $\varphi = {\overline{a}}_2 \psi a_1$, $E = T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2}$, and $F = T_{a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_1},$ holds if we impose the weaker conditions $a_1 {\mathcal {K}}_1 \subseteq {\mathcal {H}}_1$ and $a_2 {\mathcal {K}}_2 \subseteq {\mathcal {H}}_2$. However, with these weaker conditions, the operators E and F may not be invertible and so (2.15) does not provide an equivalence between $T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}$ and $T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}$ as in (1.2).

When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {H}}$, ${\mathcal {K}}_1 = {\mathcal {K}}_2 = {\mathcal {K}}$, Theorem 2.12 becomes the following.

Corollary 2.16

Let $a_1, a_2 \in {\mathscr {G}}\!L^{\infty }$ such that $a_i {\mathcal {K}}= {\mathcal {H}}$, $i = 1, 2$. Then if $\varphi , \psi \in L^{\infty }$ with $\varphi = {\overline{a}}_2\psi a_1$, we have

$$\begin{aligned} T_{\varphi }^{{\mathcal {K}}} = E T_{\psi }^{{\mathcal {H}}} F, \end{aligned}$$

(2.17)

with $E = T_{{\overline{a}}_2}^{{\mathcal {H}}, {\mathcal {K}}}$ and $F = T_{a_1}^{{\mathcal {K}}, {\mathcal {H}}} = M_{a_1}|_{{\mathcal {K}}}.$

At this point, it makes sense to ask under what conditions does (2.17) define a similarity or a unitary equivalence. Regarding similarity, we have the following.

Proposition 2.18

With the assumptions and notation as in Corollary 2.16, the following are equivalent:

• (a) the operators $T_{\varphi }^{{\mathcal {K}}}$ and $T_{\psi }^{{\mathcal {H}}}$ are similar via (2.17);

• (b) $E = F^{-1}$;

• (c) $T_{{\overline{a}}_2 - a_{1}^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} = 0$;

• (d) $T_{{\overline{a}}_2^{-1} - a_1}^{{\mathcal {K}}, {\mathcal {H}}} = 0$;

• (e) $T_{1 - {\overline{a}}_2 a_1}^{{\mathcal {K}}} = 0$;

• (f) $T_{1 - {\overline{a}}_2^{-1} a_{1}^{-1}}^{{\mathcal {H}}} = 0$.

Proof

Observe that Proposition 2.8 yields

$$\begin{aligned} E = F^{-1}&\iff T_{{\overline{a}}_2}^{{\mathcal {H}}, {\mathcal {K}}} = T_{a_{1}^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} \iff T_{{\overline{a}}_2 - a_{1}^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} = 0\\&\iff P_{{\mathcal {K}}} {\overline{a}}_2 \big (1 - a_{1}^{-1} {\overline{a}}_2^{-1}\big ) P_{{\mathcal {H}}}|_{{\mathcal {H}}} = 0\\&\iff (P_{{\mathcal {K}}} {\overline{a}}_2 P_{{\mathcal {H}}}) \Big (P_{{\mathcal {H}}} (1 - a_{1}^{-1} {\overline{a}}_2^{-1}) P_{{\mathcal {H}}}\Big )|_{{\mathcal {H}}} = 0\\&\iff E T_{1 - {\overline{a}}_2^{-1} a_{1}^{-1}}^{{\mathcal {H}}} = 0 \iff T_{1 - {\overline{a}}_2^{-1} a_{1}^{-1}}^{{\mathcal {H}}} = 0, \end{aligned}$$

where we have taken into account that, by Corollary 2.4, ${\overline{a}}_2 {\mathcal {H}}^{\perp } = {\mathcal {K}}^{\perp }$ and E is invertible. Analogously, Proposition 2.8 yields

$$\begin{aligned} E^{-1} = F \iff T_{{\overline{a}}_2^{-1} - a_1}^{{\mathcal {K}}, {\mathcal {H}}} = 0 \iff E^{-1} T_{1 - a_1 {\overline{a}}_2}^{{\mathcal {K}}} = 0 \iff T_{1 - a_{1} {\overline{a}}_2}^{{\mathcal {K}}} = 0, \end{aligned}$$

where we took into account that ${\overline{a}}_2^{-1} {\mathcal {K}}^{\perp } = {\mathcal {H}}^{\perp }$. $\square$

Remark 2.19

Naturally, the operator $F^{-1} T_{\psi }^{{\mathcal {H}}} F: {\mathcal {K}}\rightarrow {\mathcal {K}}$ is similar to $T^{{\mathcal {H}}}_{\psi }$ for any invertible $F \in {\mathcal {B}}({\mathcal {K}}, {\mathcal {H}})$. However, in general, it is not a generalized Toeplitz operator of the form $T^{{\mathcal {K}}}_{\psi }$ (see Example 6.4).

Regarding the possibility that (2.17) defines a unitary equivalence between the operators $T_{\varphi }^{{\mathcal {K}}}$ and $T_{\psi }^{{\mathcal {H}}}$, which can happen only when the multiplication operator $M_{a_1}: {\mathcal {K}}\rightarrow {\mathcal {H}}$ is unitary, we have the following.

Corollary 2.20

With the same assumptions as in Corollary 2.16, the following are equivalent:

• (a) the identity in (2.17) is a unitary equivalence;

• (b) $E = F^{-1} = F^{*}$;

• (c) $T_{1 - {\overline{a}}_2 a_1}^{{\mathcal {K}}} = 0$ and $T_{1 - |a_1|^2}^{{\mathcal {K}}} = 0$.

Proof

The condition $T_{1 - {\overline{a}}_2 a_1}^{{\mathcal {K}}} = 0$ is equivalent to $E = F^{-1}$ while the condition $T_{1 - |a_1|^2}^{{\mathcal {K}}} = 0$ is equivalent to $F^{-1} = F^{*}$. $\square$

Corollary 2.21

When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {H}}$, ${\mathcal {K}}_1 = {\mathcal {K}}_2 = {\mathcal {K}}$, $a_1 = a_2 = a \in {\mathscr {G}}\!L^{\infty }$, and $a {\mathcal {K}}= {\mathcal {H}}$, the identity in (2.17) takes the form

$$\begin{aligned} T_{\varphi }^{{\mathcal {K}}} = F^{*} T_{\psi }^{{\mathcal {H}}} F, \end{aligned}$$

with $F = T_{a}^{{\mathcal {K}}, {\mathcal {H}}} = M_{a}|_{{\mathcal {K}}}$ and $\psi = |a|^{-2} \varphi$, and $T_{\varphi }^{{\mathcal {K}}}$ is unitarily equivalent to $T_{\psi }^{{\mathcal {H}}}$ if and only if $T_{1 - |a|^2}^{{\mathcal {K}}} = 0,$ or equivalently, $T_{{\overline{a}} - a^{-1}}^{{\mathcal {H}}, {\mathcal {K}}} = 0.$ In this case, $F^{*} = F^{-1}$, making $M_{a}|_{{\mathcal {K}}}: {\mathcal {K}}\rightarrow {\mathcal {H}}$ an isometric isomorphism.

The question of similarity or unitary equivalence is therefore related to the delicate question of characterizing the functions $\eta \in L^{\infty }$ for which $T_{\eta }^{{\mathcal {H}}, {\mathcal {K}}} = 0$. For certain ${\mathcal {H}}$ and ${\mathcal {K}}$, this takes place only when the symbol is zero. For other choices of ${\mathcal {H}}$ and ${\mathcal {K}}$, however, there is often a rich variety of symbols $\eta$ corresponding to the zero generalized Toeplitz operator. We will explore this in the next several sections. Let us now discuss Theorem 2.12 in the case of the classical Toeplitz and Hankel operators as well as truncated Toeplitz operators.

2.1 Classical Toeplitz operators

Consider the classical Toeplitz operators $T_{\varphi }$ on $H^2$, where $\varphi \in L^{\infty }$ (recall the definition in (2.2)). Setting ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {K}}_1 = {\mathcal {K}}_2 = H^2$ in Theorem 2.12, we have the following equivalence.

Theorem 2.22

Let $a_1, a_2 \in {\mathscr {G}}\!H^{\infty }$ and $\varphi , \psi \in L^{\infty }$ with $\varphi = {\overline{a}}_2 \psi a_1$. Then

$$\begin{aligned} T_{\varphi } = T_{{\overline{a}}_2} T_{\psi } T_{a_1} \end{aligned}$$

(2.23)

and thus $T_{\varphi }$ is equivalent to $T_{\psi }$.

The alert reader will recognize an alternative proof of Theorem 2.22 from the well known Brown–Halmos theorem [5, Thm. 8]. One will also recognize a converse to Theorem 2.22 in that if $T_{\varphi } = T_{{\overline{a}}_2} T_{\psi } T_{a_1}$ with $a_1, a_2 \in {\mathscr {G}}\!H^{\infty }$, then $T_{\varphi } = T_{{\overline{a}}_2 \psi a_1}$ and, by the uniqueness of symbols for Toeplitz operators [20, p. 179], $\varphi = {\overline{a}}_2 \psi a_1$. In the following section we will recast this in terms of equivalence of the symbols of Toeplitz operators. The uniqueness of these symbols, and the fact that $H^{\infty } \cap \overline{H^{\infty }} = {\mathbb {C}}$, also shows that (2.23) never provides a similarity between two Toeplitz operators unless we are in the degenerate case where $a_1$ and $a_2$ are constant functions.

2.2 Classical Hankel operators

For $\varphi \in L^{\infty }$ recall the classical Hankel operator $H_{\varphi }: H^2 \rightarrow (H^2)^{\perp }$ defined by $H_{\varphi } = P_{-} \varphi P_{+}|_{H^2}$ [38, 39]. Taking ${\mathcal {H}}_1 = {\mathcal {K}}_1 = H^2$ and ${\mathcal {H}}_2 = {\mathcal {K}}_2 = (H^2)^{\perp }$ in Theorem 2.12, we have the following.

Theorem 2.24

Let $a_1, a_2 \in {\mathscr {G}}\!H^{\infty }$ and $\varphi , \psi \in L^{\infty }$ with $\varphi = a_2 \psi a_1$. Then

$$\begin{aligned} H_{\varphi } = S_{a_{2}} H_{\psi } T_{a_{1}}, \end{aligned}$$

where $S_{a_{2}} = P_{-} a_{2} P_{-}|_{(H^2)^{\perp }}$ is an invertible dual Toeplitz operator and $T_{a_{1}}$ is an invertible Toeplitz operator. Thus, $H_{\varphi }$ and $H_{\psi }$ are equivalent Hankel operators.

See [19, 42] for more on the dual of a Toeplitz operator. The non-uniqueness of symbols of Hankel operators (indeed $H_{\varphi } \equiv 0$ if and only of $\varphi \in H^{\infty }$) prevents us from formulating a converse of Theorem 2.24 as we did for Toeplitz operators. See Sect. 3 where we address this issue further.

2.3 Truncated Toeplitz operators

This section gives a version of Theorem 2.12 in the special case where the generalized Toeplitz operator becomes a truncated Toeplitz operator. For an inner function $\alpha$, we let $P_{\alpha }$ denote the orthogonal projection of $L^2$ onto ${\mathcal {K}}_{\alpha }$. For $\varphi \in L^{\infty }$ and inner functions $\alpha$ and $\gamma$, let

$$\begin{aligned} A_{\varphi }^{\gamma , \alpha }: {\mathcal {K}}_{\gamma } \rightarrow {\mathcal {K}}_{\alpha }, \quad A^{\gamma , \alpha }_{\varphi } f:= P_{\alpha }(\varphi f) \end{aligned}$$

(2.25)

be the asymmetric truncated Toeplitz operator from ${\mathcal {K}}_{\gamma }$ to ${\mathcal {K}}_{\alpha }$ with symbol $\varphi$ [6, 11]. When $\gamma = \alpha$ we write $A^{\gamma , \gamma }_{\varphi } = A^{\gamma }_{\varphi }$, which is called the truncated Toeplitz operator on ${\mathcal {K}}_{\gamma }$ with symbol $\varphi$ [40]. Under the right circumstances, one can define $A^{\gamma , \alpha }_{\varphi }$ when $\varphi \in L^2$ (define it densely on bounded functions in ${\mathcal {K}}_{\gamma }$ and consider if it has a bounded extension to ${\mathcal {K}}_{\gamma }$). Surprisingly, not every one of these operators can be represented by an $L^{\infty }$ symbol $\varphi$ [1]. However, since we will be relating equivalence of symbols with equivalence of these operators, we will focus our attention solely on bounded symbols $\varphi$. The symbol $\varphi$ for $A^{\gamma }_{\varphi }$ is not unique. Indeed [40],

$$\begin{aligned} A^{\gamma }_{\varphi } \equiv 0 \iff \varphi \in {\overline{\gamma }} \overline{H^{2}} + \gamma H^{2}. \end{aligned}$$

(2.26)

In a similar way [35],

$$\begin{aligned} A^{\gamma , \alpha }_{\varphi } \equiv 0 \iff \varphi \in {\overline{\gamma }} \overline{H^{2}} + \alpha H^{2}. \end{aligned}$$

(2.27)

We now apply our main equivalence theorem (Theorem 2.12) to these types of operators. To this end, let ${\mathcal {H}}_1 = {\mathcal {K}}_{\theta }$, ${\mathcal {H}}_2 = {\mathcal {K}}_{\alpha }$, ${\mathcal {K}}_1 = {\mathcal {K}}_{\eta }$, ${\mathcal {K}}_{2} = {\mathcal {K}}_{\gamma },$ where $\theta , \alpha , \eta$, and $\gamma$ are inner functions. Recall from Proposition 2.3 that for inner functions $\theta _1, \theta _2$ we have that $a {\mathcal {K}}_{\theta _2} = {\mathcal {K}}_{\theta _1}$ with $a \in L^{\infty }$ if and only if $a \in {\mathscr {G}}\!H^{\infty }$ and $\theta _1 = h_{-} \theta _2 a$ and $h_{-} \in {\mathscr {G}}\!\overline{H^{\infty }}$. From the identity

$$\begin{aligned} 1 = \theta _1 {\overline{\theta }}_1 = h_{-} \theta _2 a {\overline{h}}_{-} {\overline{\theta }}_{2} {\overline{a}} = h_{-} a {\overline{h}}_{-} {\overline{a}} \end{aligned}$$

we see that $h_{-} {\overline{a}} = {\overline{h}}_{-}^{-1} a^{-1} = k \in {\mathbb {C}}$ with $|k| = 1$ and so $h_{-} = k {\overline{a}}^{-1}$ and we can assume without loss of generality that

$$\begin{aligned} {\mathcal {K}}_{\theta _1} = a {\mathcal {K}}_{\theta _2} \iff \theta _{1} = {\overline{a}}^{-1} \theta _2 a \quad \hbox { with}\ a \in {\mathscr {G}}\!H^{\infty }. \end{aligned}$$

With this in mind, a version of Theorem 2.12 for truncated Toeplitz operators becomes the following.

Theorem 2.28

Let $\theta , \alpha , \eta , \gamma$ be inner functions such that

$$\begin{aligned} \theta = {\overline{a}}_{1}^{-1} \eta a_1, \quad \alpha = {\overline{a}}_{2}^{-1} \gamma a_2 \quad \text{ with }\ a_1, a_2 \in {\mathscr {G}}\!H^{\infty }. \end{aligned}$$

Then for any $\varphi \in L^{\infty }$, $A_{\varphi }^{\theta , \alpha }$ is equivalent to $A_{{\widetilde{\varphi }}}^{\eta , \gamma }$ with ${\widetilde{\varphi }} = {\overline{a}}_{2} \varphi a_1$, and we have

$$\begin{aligned} A_{\varphi }^{\theta , \alpha } = A_{{\overline{a}}_{2}^{-1}}^{\gamma , \alpha } A_{{\widetilde{\varphi }}}^{\eta , \gamma } A_{a_{1}^{-1}}^{\theta , \eta } = A_{{\overline{a}}_{2}^{-1}}^{\gamma , \alpha } A_{{\widetilde{\varphi }}}^{\eta , \gamma } M_{a_{1}^{-1}}|_{{\mathcal {K}}_{\theta }}. \end{aligned}$$

(2.29)

In particular, if $\theta = \alpha$, $\eta = \gamma$, and $a_1 = a_2 = a$, the identity in (2.29) becomes

$$\begin{aligned} A_{\varphi }^{\alpha } = A_{{\overline{a}}^{-1}}^{\gamma , \alpha } A_{{\widetilde{\varphi }}}^{\gamma } A_{a^{-1}}^{\alpha , \gamma } = A_{{\overline{a}}^{-1}}^{\gamma , \alpha } A_{{\widetilde{\varphi }}}^{\gamma } M_{a^{-1}}|_{{\mathcal {K}}_{\alpha }}, \end{aligned}$$

(2.30)

with ${\widetilde{\varphi }} = \varphi |a|^2$.

We will have more to say about truncated Toeplitz operators in §6 and we will give another reformulation of Theorem 2.12 involving dual truncated Toeplitz operators in sections Sect. 7.

3 Multipliers and generalized Toeplitz kernels and ranges

Let us keep the notation from before: ${\mathcal {H}}, {\mathcal {H}}_1, {\mathcal {H}}_2, {\mathcal {K}}, {\mathcal {K}}_1, {\mathcal {K}}_2$ are closed subspaces of $L^2$ and $\varphi , \psi \in L^{\infty }$. One of the main motivations for studying equivalence of operators of the form (1.1), with symbols $\varphi$ and $\psi$ related by a factorization $\varphi = {\overline{a}}_2 \psi a_1$ as in Theorem 2.12, stems from the study of Toeplitz operators and singular integral equations. In fact, the characterization of Toeplitz kernels and ranges, invertibility, and Fredholm properties is strongly related to the existence of an appropriate factorization of their symbols (for instance a Wiener–Hopf factorization or an almost periodic factorization). This allows us to establish an equivalence with Toeplitz operators with monomial symbols or simple exponential symbols—which are much easier to understand.

Note that when $\varphi \in L^{\infty }$ with $|\varphi | = 1$ on ${\mathbb {T}}$, we have

$$\begin{aligned} {\text {ker}}T_{\varphi } \not = \{0\} \iff \varphi = {\overline{z}} {\overline{I}} {\overline{F}} F^{-1}, \end{aligned}$$

where I is an inner function and $F \in H^2$ is outer [12, 36]. If $F \in {\mathscr {G}}\!H^{\infty }$, then for $\theta = z I$ we have that $\varphi = {\overline{F}} {\overline{\theta }} F^{-1}$ and, by Theorem 2.12, $T_{\varphi }$ is equivalent to $T_{{\overline{\theta }}}$, with

$$\begin{aligned} {\text {ker}}T_{\varphi } = F {\text {ker}}T_{{\overline{\theta }}} = F {\mathcal {K}}_{\theta }. \end{aligned}$$

This is true because if $g_{+} \in {\mathscr {G}}\!H^{\infty }$ and $g_{-} \in {\mathscr {G}}\!\overline{H^{\infty }}$, then for any ${\widetilde{\varphi }} \in L^{\infty }$,

$$\begin{aligned} {\text {ker}}T_{{\widetilde{\varphi }} g_{+}} = g_{+}^{-1} {\text {ker}}T_{{\widetilde{\varphi }}} \quad \text{ and } \quad {\text {ker}}T_{{\widetilde{\varphi }} g_{-}} = {\text {ker}}T_{{\widetilde{\varphi }}}. \end{aligned}$$

(3.1)

One may ask about the relationship between the kernels of two Toeplitz operators when the symbol of one of them is multiplied by another function in ${\mathscr {G}}\!L^{\infty }$ which is not necessarily in ${\mathscr {G}}\!H^{\infty }$ or ${\mathscr {G}}\!\overline{H^{\infty }}$. If that function is a non-constant inner function, which also belongs to $H^{\infty } {\setminus } {\mathscr {G}}\!H^{\infty }$, we have that

$$\begin{aligned} {\text {ker}}T_{{\widetilde{\varphi }} \theta } \subsetneq {\text {ker}}T_{{\widetilde{\varphi }}} \quad \text{ and } \quad \theta {\text {ker}}T_{{\widetilde{\varphi }} \theta } \subsetneq {\text {ker}}T_{{\widetilde{\varphi }}}. \end{aligned}$$

(3.2)

If we now consider Hankel operators instead, we see that with the same notation, if $g_{+} \in {\mathscr {G}}\!H^{\infty }$ and $\theta$ is inner,

$$\begin{aligned}{} & {} {\text {ker}}H_{{\widetilde{\varphi }} g_{+}} = {\text {ker}}H_{{\widetilde{\varphi }}}, \end{aligned}$$

(3.3)

$$\begin{aligned}{} & {} {\text {ker}}H_{{\widetilde{\varphi }} \theta } \supseteq {\text {ker}}H_{{\widetilde{\varphi }}} \quad \text{ and } \quad \theta {\text {ker}}H_{{\widetilde{\varphi }} \theta } \subseteq {\text {ker}}H_{{\widetilde{\varphi }}}, \end{aligned}$$

(3.4)

while no simple relation corresponds to the second equality in (3.1) for Hankel operators.

In this section we show that by interpreting the relations (3.1) through (3.4) in terms of multipliers, we can put them in a general context and, along the way, extend them to generalized Toeplitz operators.

If two Hilbert space operators A and B are equivalent via (1.2), with $A = E B F$, where E and F are invertible operators, it is clear that their kernels and ranges are isomorphic with ${\text {ker}}A = F^{-1} {\text {ker}}B$ and ${\text {Ran}} A = E {\text {Ran}} B.$ More generally we have the following.

Proposition 3.5

Let $A \in {\mathscr {B}}({\mathcal {K}}_1, {\mathcal {K}}_2)$ and $B \in {\mathscr {B}}({\mathcal {H}}_1, {\mathcal {H}}_2)$ with $A = E B F$, where $E \in {\mathscr {B}}({\mathcal {H}}_2, {\mathcal {K}}_2)$ and $F \in {\mathscr {B}}({\mathcal {K}}_1, {\mathcal {H}}_1)$.

• (a) If E is invertible, then $F {\text {ker}}A \subseteq {\text {ker}}B$ and $E {\text {Ran}} B \supseteq {\text {Ran}} A.$

• (b) If F is invertible, then ${\text {ker}}B \subseteq F {\text {ker}}A$ and $E {\text {Ran}} B \subseteq {\text {Ran}} A.$

Proof

For the proof of (a), note that

$$\begin{aligned} A f = 0 \iff E B F f = 0 \iff B F f = 0 \implies F f \in {\text {ker}}B. \end{aligned}$$

Moreover,

$$\begin{aligned} A f = g \iff E B F f = g \iff B F f = E^{-1} g \iff E^{-1} g \in {\text {Ran}} B. \end{aligned}$$

For the proof of (b), note that

$$\begin{aligned} B f = 0&\iff B F (F^{-1} f) = 0\\&\implies E B F(F^{-1} f) = 0\\&\iff A(F^{-1} f) = 0\\&\implies F^{-1} f \in {\text {ker}}A. \end{aligned}$$

Moreover, $B f = g \implies E B f = E g \iff E B F (F^{-1} f) = E g$ $\square$

Concerning the case where the operators A and B are generalized Toeplitz operators, one can take into account Theorem 2.7, Remark 2.14, and Proposition 3.5 to formulate the following.

Proposition 3.6

Let $a_1, a_2 \in L^{\infty }$ with $a_1 {\mathcal {K}}_1 \subseteq {\mathcal {H}}_1$, $a_2 {\mathcal {K}}_2 \subseteq {\mathcal {H}}_2$, and let $\varphi = {\overline{a}}_2 \psi a_1$.

• (a) If $a_2 \in {\mathscr {G}}\!L^{\infty }$ and $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$, then

  $$\begin{aligned} a_1 {\text {ker}}T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} \subseteq {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} \quad \text{ and } \quad T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} {\text {Ran}}(T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}) \supseteq {\text {Ran}}(T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}). \end{aligned}$$

• (b) If $a_1 \in {\mathscr {G}}\!L^{\infty }$ and $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$, then

  $$\begin{aligned} {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} \subseteq a_1 {\text {ker}}_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} \quad \text{ and } \quad T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} {\text {Ran}}(T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}) \subseteq {\text {Ran}}(T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}). \end{aligned}$$

Some immediate consequences of Proposition 3.6 now follow.

Corollary 3.7

Let $a_1, a_2 \in L^{\infty }$.

• (a) If $a_1 {\mathcal {K}}_1 \subseteq {\mathcal {H}}_1$ we have

  $$\begin{aligned} a_1 {\text {ker}}T_{\psi a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_2} \subseteq {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} \; \; \text{ and } \; \; {\text {Ran}}(T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}) \supseteq {\text {Ran}}(T_{\psi a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_2}). \end{aligned}$$

  (3.8)

• (b) If $a_2 {\mathcal {K}}_2 \subseteq {\mathcal {H}}_2$ we have

  $$\begin{aligned} {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} \subseteq {\text {ker}}T_{{\overline{a}}_2 \psi }^{{\mathcal {H}}_1, {\mathcal {K}}_2} \; \; \text{ and } \; \; T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} {\text {Ran}}(T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}) \subseteq {\text {Ran}}(T_{{\overline{a}}_2 \psi }^{{\mathcal {H}}_1, {\mathcal {K}}_2}).\qquad \quad \end{aligned}$$

  (3.9)

• (c) If $a_1, a_2 \in {\mathscr {G}}\!L^{\infty }$ with $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$, $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$, and $\varphi = {\overline{a}}_2 \psi a_1$, then

  $$\begin{aligned} a_1 {\text {ker}}T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} \; \text{ and } \; {\text {Ran}}(T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2}) = T_{{\overline{a}}_2}^{{\mathcal {H}}_2, {\mathcal {K}}_2} {\text {Ran}}(T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}).\qquad \qquad \end{aligned}$$

  (3.10)

Corollary 3.11

If $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$, then for any ${\mathcal {H}}_2$

$$\begin{aligned} a_{1}^{-1} {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} = {\text {ker}}T_{\psi a_{1}}^{{\mathcal {K}}_1, {\mathcal {H}}_2}. \end{aligned}$$

(3.12)

If $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$, then for any ${\mathcal {H}}_1$,

$$\begin{aligned} {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2} = {\text {ker}}T_{{\overline{a}}_2 \psi }^{{\mathcal {H}}_1, {\mathcal {K}}_2}. \end{aligned}$$

(3.13)

Remark 3.14

• (a) For Toeplitz operators on $H^2$, the equalities from (3.1) can be seen as particular cases of (3.12) and (3.13), while the inclusion in (3.2) can be see as a particular case of (3.9), with $a_2 = \theta$ and $\psi = {\widetilde{\varphi }} \theta$.

• (b) For Hankel operators, the equality (3.3) is a particular case of (3.13), while the first inclusion in (3.4) follows from (3.9), with $a_2 = {\overline{\theta }}$ and $\psi = {\widetilde{\varphi }}$, and the second inclusion follows from (3.8).

• (c) From the previous results we see that if $a_1$ is a multiplier from ${\mathcal {K}}_1$ into (onto) ${\mathcal {H}}_1$, then $a_1$ is also a multiplier from ${\text {ker}}T_{{\overline{a}}_2 \psi a_1}^{{\mathcal {K}}_1, {\mathcal {K}}_2}$ into (onto) ${\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}$ if $a_2 \in {\mathscr {G}}\!L^{\infty }$, $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$. In particular, $a_1$ is a multiplier from ${\text {ker}}T_{\psi a_1}^{{\mathcal {K}}_1, {\mathcal {H}}_2}$ into (onto) ${\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}$.

For Toeplitz operators on $H^2$, if the first equality in (3.10) holds for some $a_1 \in {\mathscr {G}}\!H^{\infty }$, then we must have $\varphi = {\overline{a}}_2 \psi a_1$ for some $a_2 \in {\mathscr {G}}\!H^{\infty }$, i.e.,

$$\begin{aligned} {\text {ker}}T_{\varphi } = a_{1}^{-1} {\text {ker}}T_{\psi } \implies \varphi = {\overline{a}}_2 \psi a_1 \quad \hbox { for some}\ a_2 \in {\mathscr {G}}\!H^{\infty } \end{aligned}$$

by [12, Prop. 2.16] (because in this case we have ${\text {ker}}T_{ \varphi } = {\text {ker}}T_{a_1 \psi }$). It is therefore natural to ask if, for generalized Toeplitz operators, some sort of converse to Corollary 3.7(c) is true. Namely, if (3.10) holds for some $a_1 \in {\mathscr {G}}\!L^{\infty }$ with $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$, is it the case that the symbols $\varphi$ and $\psi$ are related by $\varphi = {\overline{a}}_2 \psi a_1$ for some $a_2 \in {\mathscr {G}}\!L^{\infty }$ with $a_2 {\mathcal {K}}_2 = {\mathcal {H}}_2$? This is not quite true. However, from Corollary 3.11, we do have the following.

Proposition 3.15

Let $a_1 \in {\mathscr {G}}\!L^{\infty }$ with $a_1 {\mathcal {K}}_1 = {\mathcal {H}}_1$. If

$$\begin{aligned} {\text {ker}}T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = a_{1}^{-1} {\text {ker}}T_{\psi }^{{\mathcal {H}}_1, {\mathcal {H}}_2}, \end{aligned}$$

then there exists ${\widetilde{\varphi }} \in L^{\infty }$ such that

$$\begin{aligned} {\text {ker}}T_{\varphi }^{{\mathcal {K}}_1, {\mathcal {K}}_2} = {\text {ker}}T_{{\widetilde{\varphi }}}^{{\mathcal {K}}_1, {\mathcal {K}}_2} \end{aligned}$$

and ${\widetilde{\varphi }} = {\overline{a}}_{2} \psi a_1$ for any $a_2$ such that $a_2 {\mathcal {H}}_2 = {\mathcal {K}}_2$.

Note how this gives somewhat of a converse to Theorem 2.12.

4 Complex selfadjointness

For a closed subspace ${\mathcal {K}}$ of $L^2$, consider a mapping $C_{{\mathcal {K}}}: {\mathcal {K}}\rightarrow {\mathcal {K}}$ for which $C_{{\mathcal {K}}}(f + a g) = C_{{\mathcal {K}}} f + {\overline{a}} C_{{\mathcal {K}}} g$ for all $f, g, \in {\mathcal {K}}$ and $a \in {\mathbb {C}}$. Such a mapping is antilinear. For an antilinear mapping $C_{{\mathcal {K}}}$, one can define an antilinear adjoint $C_{{\mathcal {K}}}^{*}: {\mathcal {K}}\rightarrow {\mathcal {K}}$ which satisfies $\langle C_{{\mathcal {K}}} f, g\rangle = \overline{\langle f, C^{*}_{{\mathcal {K}}} g\rangle }$ for all $f, g \in {\mathcal {K}}$. The mapping $C_{{\mathcal {K}}}$ is called an antilinear unitary mapping if $C_{{\mathcal {K}}}^{*} = C_{{\mathcal {K}}}^{-1}$. We will give some examples in a moment.

Remark 4.1

In many settings, one imposes the additional condition that $C_{{\mathcal {K}}}^2 = I_{{\text {d}}}$ and such involutive and antilinear unitary mappings are called conjugations. Many of the $C_{{\mathcal {K}}}$ we will present below will be involutive. However, this extra hypothesis is not necessary for our results and so we leave it out. The version without the involutive assumption was explored in [9] while the version with the involutive assumption was explored in [26,27,28].

Let $C_{{\mathcal {H}}}$ denote an antilinear unitary mapping on ${\mathcal {H}}$. For $a \in {\mathscr {G}} L^{\infty }$ with $a {\mathcal {K}}= {\mathcal {H}}$, this next result shows that, under certain conditions, the operators

$$\begin{aligned} F = M_{a}|_{{\mathcal {K}}} \quad \text{ and } \quad E = F^{*} = P_{{\mathcal {K}}} {\overline{a}} P_{{\mathcal {H}}}|_{{\mathcal {H}}} \end{aligned}$$

from Proposition 2.8 induce a corresponding antilinear unitary mapping $C_{{\mathcal {K}}}$ on ${\mathcal {K}}$.

Proposition 4.2

Let ${\mathcal {H}}$ and ${\mathcal {K}}$ be closed subspaces of $L^2$ and $C_{{\mathcal {H}}}$ be an antilinear unitary mapping on ${\mathcal {H}}$, F be an invertible operator from ${\mathcal {K}}$ onto ${\mathcal {H}}$, and $E = F^{*}$. Then for

$$\begin{aligned} C_{{\mathcal {K}}} = F^{-1} C_{{\mathcal {H}}} F, \end{aligned}$$

(4.3)

the following are equivalent:

• (a) $C_{{\mathcal {K}}}$ is an antilinear unitary mapping on ${\mathcal {K}}$;

• (b) $C_{{\mathcal {H}}} (F E) = (F E) C_{{\mathcal {H}}}$.

• (c) $C_{{\mathcal {K}}} = E C_{{\mathcal {H}}} E^{-1}$.

Proof

With $C_{{\mathcal {K}}}$ defined to be $F^{-1} C_{{\mathcal {H}}} F$, we see that $C_{{\mathcal {K}}}$ is antilinear and invertible with $C_{{\mathcal {K}}}^{*} = F^{*} C^{*}_{{\mathcal {H}}} (F^{-1})^{*} = F^{*} C_{{\mathcal {H}}}^{-1} (F^{-1})^{*}$ and $C_{{\mathcal {H}}}^{-1} = F^{-1} C_{{\mathcal {K}}}^{-1} F$. Moreover,

$$\begin{aligned} C_{{\mathcal {K}}}^{*} = C_{{\mathcal {K}}}^{-1}&\iff F^{*} C_{{\mathcal {H}}}^{-1} (F^{-1})^{*} = F^{-1} C_{{\mathcal {H}}}^{-1} F\\&\iff F F^{*} C_{{\mathcal {H}}}^{-1} (F F^{*})^{-1} = C_{{\mathcal {H}}}^{-1}\\&\iff (F F^{*}) C_{{\mathcal {H}}} = C_{{\mathcal {H}}} (F F^{*})\\&\iff (FE) C_{{\mathcal {H}}} = C_{{\mathcal {H}}} (FE)\\&\iff E C_{{\mathcal {H}}} E^{-1} = F^{-1} C_{{\mathcal {H}}} F. \end{aligned}$$

$\square$

Recall from Remark 4.1 that our antilinear isometry $C_{{\mathcal {K}}}$ is called a conjugation if $C_{{\mathcal {K}}}^2 = I_{{\text {d}}}$. Note that a conjugation satisfies $C = C^{-1}$.

Proposition 4.4

Let $C_{{\mathcal {K}}}$ be defined as in (4.3). Then $C_{{\mathcal {K}}}^2 = I_{{\text {d}}} \iff C_{{\mathcal {H}}}^2 = I_{{\text {d}}}$.

An operator T on ${\mathcal {H}}$ is complex selfadjoint with respect to an antilinear isometry C on ${\mathcal {H}}$, if $C T C^{-1} = T^{*}$ [9]. If the antilinear unitary mapping also satisfies the involutive criterion $C^2 = I_{d}$, then T is called a C-symmetric operator. These types of operators were explored in great detail in [26,27,28].

As we will see shortly, some generalized Toeplitz operators are complex self adjoint. The main theorem of this section relates the complex self adjointness of two equivalent generalized Toeplitz operators.

Theorem 4.5

For closed subspaces ${\mathcal {H}}$ and ${\mathcal {K}}$ of $L^2$ and $\varphi , \psi \in L^{\infty }$, suppose that $T_{\varphi }^{{\mathcal {K}}}$ and $T_{\psi }^{{\mathcal {H}}}$ are equivalent via

$$\begin{aligned} E = T_{{\overline{a}}}^{{\mathcal {H}}, {\mathcal {K}}} \quad \text{ and } \quad F = T_{a}^{{\mathcal {K}}, {\mathcal {H}}}, \end{aligned}$$

where $a \in {\mathscr {G}} L^{\infty }$ with $a {\mathcal {K}}= {\mathcal {H}}$. Also suppose that $C_{{\mathcal {H}}}$ is an antilinear unitary mapping on ${\mathcal {H}}$ such that $F^{-1} C_{{\mathcal {H}}} F = E C_{{\mathcal {H}}} E^{-1}$. Then $T_{\psi }^{{\mathcal {H}}}$ is complex selfadjoint with respect to $C_{{\mathcal {H}}}$ if and only if $T_{\varphi }^{{\mathcal {K}}}$ is complex selfadjoint with respect to the antilinear unitary mapping $C_{{\mathcal {K}}} = F^{-1} C_{{\mathcal {H}}} F$.

Proof

Note that

$$\begin{aligned} C_{{\mathcal {K}}} T_{\varphi }^{{\mathcal {K}}} C_{{\mathcal {K}}}^{-1} = (T_{\varphi }^{{\mathcal {K}}})^{*}&\iff (E C_{{\mathcal {H}}} E^{-1})(E T_{\psi }^{{\mathcal {H}}} F)(F^{-1} C_{{\mathcal {H}}}^{-1} F) = E (T_{\psi }^{{\mathcal {H}}})^{*} F\\&\iff C_{{\mathcal {H}}} T_{\psi }^{{\mathcal {H}}} C_{{\mathcal {H}}}^{-1} = (T_{\psi }^{{\mathcal {H}}})^{*} \end{aligned}$$

$\square$

It is natural to also consider a conjugation on ${\mathcal {K}}$ which is equivalent, via the operators E and F defined in Theorem 4.5, to the antilinear unitary mapping $C_{{\mathcal {H}}}$ on ${\mathcal {H}}$.

Proposition 4.6

With the same assumptions as in Proposition 4.2, and with

$$\begin{aligned} {\widetilde{C}}_{{\mathcal {K}}} = E C_{{\mathcal {H}}} F, \end{aligned}$$

the following are equivalent:

• (a) ${\widetilde{C}}_{{\mathcal {K}}}$ is an antilinear unitary mapping on ${\mathcal {K}}$;

• (b) $(FE) C_{{\mathcal {H}}} (FE) = C_{{\mathcal {H}}}$;

• (c) $E C_{{\mathcal {H}}} F = F^{-1} C_{{\mathcal {H}}} E^{-1}$.

Proposition 4.7

Let ${\widetilde{C}}_{{\mathcal {K}}}$ be defined as in Proposition 4.6 and assume that it is an anti-linear unitary mapping. Then $C_{{\mathcal {H}}}^2 = I_{{\text {d}}}$ if and only if ${\widetilde{C}}_{{\mathcal {K}}}^{2} = I_{{\text {d}}}$.

Proof

Notice that

$$\begin{aligned} {\widetilde{C}}_{{\mathcal {K}}}^2&= E C_{{\mathcal {H}}} (FE C_{{\mathcal {H}}}) F = E C_{{\mathcal {H}}}^2 E^{-1} F^{-1} F = I_{{\text {d}}} . \end{aligned}$$

$\square$

Note that, even if ${\widetilde{C}}_{{\mathcal {K}}}$, defined by Proposition , is an antilinear unitary mapping, we are unable to conclude, in general, that $T_{\varphi }^{{\mathcal {K}}}$ is complex selfadjoint with respect to ${\widetilde{C}}_{{\mathcal {K}}}$ when $T_{\psi }^{{\mathcal {H}}}$ is complex selfadjoint with respect to $C_{{\mathcal {H}}}$. Indeed, in general, if F is not a unitary operator, we have

$$\begin{aligned} {\widetilde{C}}_{{\mathcal {K}}} T_{\varphi }^{{\mathcal {K}}} {\widetilde{C}}^{-1}_{{\mathcal {K}}}&= F^{-1} C_{{\mathcal {H}}} E^{-1} (E T_{\psi }^{{\mathcal {H}}} F) F^{-1} C_{{\mathcal {H}}}^{-1} E^{-1}\\&= F^{-1} (T_{\psi }^{{\mathcal {H}}})^{*} E^{-1}\\&\not = F^{*} (T_{\psi }^{{\mathcal {H}}})^{*} E^{*} = (T_{\varphi }^{{\mathcal {K}}})^{*}. \end{aligned}$$

When F is a unitary operator, we have the following.

Corollary 4.8

Let $C_{{\mathcal {K}}}$ and ${\widetilde{C}}_{{\mathcal {K}}}$ be defined as in Propositions 4.2 and 4.6, respectively. If F is unitary then

• (a) $C_{{\mathcal {K}}}$ is an antilinear unitary mapping.

• (b) $C_{{\mathcal {K}}} = {\widetilde{C}}_{{\mathcal {K}}}$,

• (c) $T_{\varphi }^{{\mathcal {K}}} = E T_{\psi }^{{\mathcal {H}}} F$ is complex selfadjoint with respect to $C_{{\mathcal {K}}}$ if and only if $T_{\psi }^{{\mathcal {H}}}$ is complex selfadjoint with respect to $C_{{\mathcal {H}}}$.

Example 4.9

For an inner function $\gamma$, the truncated Toeplitz operator $A_{\varphi }^{\gamma }$ (defined in (2.25)) is complex selfadjoint [27] (see also [24, p. 291]) with respect to the conjugation $C_{\gamma }$ on the model space ${\mathcal {K}}_{\gamma }$ defined by

$$\begin{aligned} (C_{\gamma } f)(\xi ) = \gamma (\xi ) \overline{\xi f(\xi )}. \end{aligned}$$

Now let $\alpha$ be any inner function and $a \in {\mathscr {G}}\!H^{\infty }$ such that $a {\mathcal {K}}_{\alpha } = {\mathcal {K}}_{\gamma }$, i.e., $a = {\overline{\alpha }} \gamma {\overline{h}}$ for some $h \in {\mathscr {G}}\!H^{\infty }$. Since $\alpha , \gamma$ are inner, we have $|a| = |h|$ and that $a {\overline{a}} = h {{\overline{h}}}$ if and only if $a h^{-1} = {\overline{h}} {\overline{a}}^{-1}$. Since the left hand side of the last equality belongs to $H^{\infty }$ while the right hand side belongs to $\overline{H^{\infty }}$, we conclude that $a h^{-1} = {\overline{h}} {\overline{a}}^{-1}$ is a constant and therefore $h = c a$. We can assume that $c = 1$ and it follows that

$$\begin{aligned} \gamma = {\overline{a}}^{-1} \alpha a. \end{aligned}$$

(4.10)

Therefore, Theorem 2.28 says that

$$\begin{aligned} A_{|a|^2 \psi }^{\alpha } = A_{{\overline{a}}}^{\gamma , \alpha } A_{\psi }^{\gamma } A_{a}^{\alpha , \gamma }. \end{aligned}$$

Theorem 4.5 now says that $A_{|a|^2 \psi }^{\alpha }$ is complex symmetric with respect to the conjugation ${\widetilde{C}}$ defined on ${\mathcal {K}}_{\alpha }$ by ${\widetilde{C}} f = a^{-1} C_{\gamma } a f$. Note that ${\widetilde{C}}$ satisfies the equivalent conditions of Proposition 4.2. Indeed, using (4.10), we see that for any $f \in {\mathcal {K}}_{\alpha }$

$$\begin{aligned} a^{-1} C_{\gamma } a f&= a^{-1} \gamma {\overline{z}} {\overline{a}} {\overline{f}}\\&= a^{-1} ({\overline{a}}^{-1} \alpha a) {\overline{z}} {\overline{a}} {\overline{f}}\\&= \alpha {\overline{z}} {\overline{f}} \end{aligned}$$

and

$$\begin{aligned} A_{{\overline{a}}}^{\gamma , \alpha } C_{\gamma } A_{{\overline{a}}^{-1}}^{\alpha , \gamma } f&= P_{\alpha } {\overline{a}} P_{\gamma } C_{\gamma }P_{\gamma } {\overline{a}}^{-1} P_{\alpha } f\\&= P_{\alpha } {\overline{a}} P_{\gamma } C_{\gamma }{\overline{a}}^{-1} f\\&= P_{\alpha } {\overline{a}} P_{\gamma }\gamma {\overline{z}} a^{-1} {\overline{f}}\\&= P_{\alpha } {\overline{a}} P_{\gamma } {\overline{a}}^{-1} \alpha {\overline{z}} {\overline{f}}\\&= P_{\alpha } {\overline{a}} {\overline{a}}^{-1} \alpha {\overline{z}} {\overline{f}}\\&= \alpha {\overline{z}} {\overline{f}} \end{aligned}$$

and we verify that ${\widetilde{C}}$ turns out to be the usual conjugation on ${\mathcal {K}}_{\alpha }$.

5 Kernels of Toeplitz operators

When ${\mathcal {H}}_1 = {\mathcal {H}}_2 = {\mathcal {K}}_1 = {\mathcal {K}}_2 = H^2$, then $a H^2 = H^2$ if and only if $a \in {\mathscr {G}}\!H^{\infty }$. We say that $\varphi , \psi \in L^{\infty }$ are equivalent, and write $\varphi \sim \psi$, if and only if there are $a_{+} \in {\mathscr {G}}\!H^{\infty }$ and $a_{-} \in {\mathscr {G}}\!\overline{H^{\infty }}$ such that

$$\begin{aligned} \varphi = a_{-} \psi a_{+}. \end{aligned}$$

(5.1)

In this case, the Toeplitz operators $T_{\varphi }$ and $T_{\psi }$ are equivalent (see Theorem 2.22). Conversely, if $T_{\varphi }$ and $T_{\psi }$ are equivalent via (2.13), then by Corollary 3.7 there exists an $a_{+} \in {\mathscr {G}}\!H^{\infty }$ such that ${\text {ker}}T_{\psi } = a_{+} {\text {ker}}T_{\varphi }$. It follows that $\psi$ and $\varphi$ satisfy a relation of the form (5.1), so that $\varphi \sim \psi$ (see the remarks before Proposition 3.15).

It is often the case that the function $\psi$ in (5.1) is the conjugate of an inner function $\theta$, i.e., $\varphi \in {\mathscr {G}}\!L^{\infty }$ and

$$\begin{aligned} \varphi = a_{-} {\overline{\theta }} a_{+}. \end{aligned}$$

(5.2)

This is the case when the function $\varphi$ is an invertible Hölder continuous function on ${\mathbb {T}}$ with exponent $\mu \in (0, 1)$ which has a negative index with respect to the origin [37, Ch. 3, Cor. 5,2] and we take $\theta = z^n$. It is also the case for certain classes of almost periodic functions $\varphi$ on ${\mathbb {R}}$, where the inner function $\theta$ is an exponential of the form $\theta (x) = e^{i a x}$, $a \in (0, \infty )$ [3].

Let us give a few examples of equivalent functions.

Example 5.3

(Finite Blaschke products) Suppose $a_1, \ldots , a_n$ are points in the open unit disk ${\mathbb {D}}$ (repetitions allowed) and

$$\begin{aligned} \alpha (z) = \prod _{j = 1}^{n} \frac{z - a_j}{1 - {\overline{a}}_j z} \end{aligned}$$

is a finite Blaschke product [25]. Note that $\alpha$ is an inner function. For any $z \in {\mathbb {T}}$ we have the identity

$$\begin{aligned} \alpha (z) = \frac{1}{z^n} \prod _{j = 1}^{n} (z - a_j) \cdot z^n \cdot \prod _{j = 1}^{n} \frac{1}{1 - {\overline{a}}_j z} = \alpha _{-}(z) \cdot z^n \cdot \alpha _{+}(z), \end{aligned}$$

where $\alpha _{+}, {\overline{\alpha }}_{-} \in {\mathscr {G}}\!H^{\infty }$, and hence $\alpha \sim z^n$. As an immediate consequence, we see that all finite Blaschke products of the same degree (same number of zeros—counting multiplicity) are equivalent.

Example 5.4

(Generalized Frostman shifts) Let $h \in H^{\infty }$ with $\Vert h\Vert _{\infty } < 1$. For any inner function $\theta$ we define

$$\begin{aligned} \theta _{{\overline{h}}}:= \frac{\theta - {\overline{h}}}{1 - h \theta }. \end{aligned}$$

(5.5)

Using the fact that $\theta$ is inner, and thus $\theta {\overline{\theta }} = 1$ almost everywhere on ${\mathbb {T}}$, it follows that $\theta _{{\overline{h}}} \in L^{\infty }$ and $|\theta _{{\overline{h}}}| = 1$ almost everywhere on ${\mathbb {T}}$. Furthermore, a calculation shows that

$$\begin{aligned} \theta _{{\overline{h}}} = a_{-} \theta a_+, \end{aligned}$$

where

$$\begin{aligned} a_{-} = 1 - {\overline{h}} {\overline{\theta }} \in {\mathscr {G}}\!\overline{H^{\infty }} \quad \text{ and } \quad a_+ = (1 - h \theta )^{-1} \in {\mathscr {G}}\!H^{\infty }. \end{aligned}$$

Thus $\theta \sim \theta _{{\overline{h}}}$. Observe that $\theta$ is inner while $\theta _{{\overline{h}}}$ is unimodular on ${\mathbb {T}}$, but not necessarily inner (since it may not be the boundary values of a bounded analytic function on ${\mathbb {D}}$). An interesting case arises when ${\overline{h}}$ is a constant function equal to $a \in {\mathbb {D}}$. In this case, the function $\theta _{{\overline{h}}}$ from (5.5) becomes

$$\begin{aligned} \theta _{a}(z) = \frac{\theta (z) - a}{1 - {\overline{a}} \theta (z)}, \end{aligned}$$

(5.6)

which is a Frostman shift of $\theta$ (and also an inner function). Thus $\theta$ and $\theta _a$ are equivalent inner functions for every $a \in {\mathbb {D}}$. A classical result of Frostman [23] (see also [29, p. 75]) says that $\theta _a$ is a Blaschke product for “most” $a \in {\mathbb {D}}$.

It follows from (5.2) that $a_{+}^{-1}$ is a multiplier from ${\mathcal {K}}_{\theta }$ onto ${\text {ker}}T_{\varphi }$, thus yielding two isomorphisms between ${\text {ker}}T_{\varphi }$ and ${\mathcal {K}}_{\theta }$ as in Proposition 2.8. This gives us the following (taking Proposition 2.4 and Corollary 3.7 into account).

Proposition 5.7

If $\varphi = a_{-} {\overline{\theta }} a_{+}$, where $a_{+}, {\overline{a}}_{-} \in {\mathscr {G}}\!\overline{H^{\infty }}$, then

• (a) ${\text {ker}}T_{\varphi } = a_{+}^{-1} {\mathcal {K}}_{\theta }$.

• (b) $({\text {ker}}T_{\varphi })^{\perp } = {\overline{a}}_{+} {\mathcal {K}}_{\theta }^{\perp }$.

Moreover, if (5.2) holds, when studying ${\text {ker}}T_{\varphi }$ we can assume, without loss of generality, that $|\varphi | = 1$ almost everywhere on ${\mathbb {T}}$ because ${\text {ker}}T_{\varphi } = {\text {ker}}T_{{\widetilde{\varphi }}}$ with ${\widetilde{\varphi }} = {\overline{a}}_{+} {\overline{\theta }} a_{+}$ (since $a_{-}, {\overline{a}}_{+} \in {\mathscr {G}}\!\overline{H^{\infty }}$ [12]). On the other hand, if $|\varphi | = 1$ almost everywhere on ${\mathbb {T}}$ and $\varphi = a_{-} {\overline{\theta }} a_{+}$, it is not difficult to see that $a_{-}$ and ${\overline{a}}_{+}^{-1}$ must differ by a multiplicative constant. Thus, if $|\varphi | = 1$ almost everywhere on ${\mathbb {T}}$ and (5.2) holds, then

$$\begin{aligned} ({\text {ker}}T_{\varphi })^{\perp } = a_{-}^{-1} {\mathcal {K}}_{\theta }^{\perp }. \end{aligned}$$

The equality in Proposition 5.7 has a connection with Hayashi’s representation of Toeplitz kernels. In [32], Hayashi proved that the kernel K of a Toeplitz operator with symbol $\varphi$ can be written as $K = f_{+} {\mathcal {K}}_{\alpha }$, where, in particular, $f_{+}$ is outer, $\alpha$ is inner with $\alpha (0) = 0$, and $f_{+}$ multiplies the model space ${\mathcal {K}}_{\alpha }$ isometrically onto K. When $f_{+} \in {\mathscr {G}}\!H^{\infty }$, we must have $\varphi = f_{-} {\overline{\alpha }} f_{+}^{-1}$ for some $f_{-} \in {\mathscr {G}}\!\overline{H^{\infty }}$ (see Sect. 3). In general, neither the outer function $f_{+}$ nor the inner function $\alpha$ in Hayashi’s representation of a Toeplitz kernel are explicitly known. However, the representation in Proposition 5.7 is explicit whenever one can factor $\varphi$ as in (5.2). This can be done for a wide class of functions and provides an isomorphism between ${\text {ker}}T_{\varphi }$ and ${\mathcal {K}}_{\theta }$ through an invertible bounded multiplier. From here, one might ask when is this isomorphism isometric, as in Hayashi’s representation. The answer lies in the relationship between the two isomorphisms from ${\text {ker}}T_{\varphi }$ onto ${\mathcal {K}}_{\theta }$, namely

$$\begin{aligned} I_{1}:= M_{a_{+}}|_{{\text {ker}}T_{\varphi }} \quad \text{ and } \quad I_{2}:= P_{\theta } {\overline{a}}_{+}^{-1} I_{{\text {d}}}|_{{\text {ker}}T_{\varphi }} = (I_{1}^{-1})^{*}, \end{aligned}$$

and, by Corollary 2.21, can be given in terms of a truncated Toeplitz operator as follows.

Proposition 5.8

Let $\varphi \sim {{\overline{\theta }}}$ with $\varphi = a_{-} {\overline{\theta }} a_{+}$, where $a_{+}, {\overline{a}}_{-} \in {\mathscr {G}}\!H^{\infty }$ and $\theta$ is inner. Then $M_{a_{+}^{-1}}: {\mathcal {K}}_{\theta } \rightarrow {\text {ker}}T_{\varphi }$ is an isometric isomorphism if and only if

$$\begin{aligned} A^{\theta }_{1 - |a_{+}|^{-2}} = 0 \end{aligned}$$

or, equivalently,

$$\begin{aligned} 1 - |a_{+}|^{-2} \in \overline{\theta H^2} + \theta H^2. \end{aligned}$$

Example 5.9

(Generalized Crofoot transform) For $h \in H^{\infty }$ with $\Vert h\Vert _{\infty } < 1$ and $\theta$ inner, recall

$$\begin{aligned} \theta _{{\overline{h}}} = \frac{\theta - {\overline{h}}}{1 - h \theta }, \end{aligned}$$

the generalized Frostman shift from Example 5.4. For any $k \in {\mathbb {C}}\setminus \{0\}$,

$$\begin{aligned} \overline{\theta _{{\overline{h}}}} = a_{-} {\overline{\theta }} a_{+}, \end{aligned}$$

where

$$\begin{aligned} a_{+}:= k^{-1} (1 - h \theta ) \in {\mathscr {G}}\!H^{\infty } \quad \text{ and } \quad a_{-} = k (1 - {\overline{h}} {\overline{\theta }})^{-1} \in {\mathscr {G}}\!\overline{H^{\infty }}. \end{aligned}$$

From Proposition 5.7 we know that

$$\begin{aligned} J_{{\overline{h}}}:= \frac{k}{1 - h \theta } \end{aligned}$$

is a multiplier from ${\mathcal {K}}_{\theta }$ onto ${\text {ker}}T_{\overline{\theta _{{\overline{h}}}}}$. We call the operator of multiplication by $J_{{\overline{h}}}$ a generalization of the Crofoot transform, for it includes, as a special case, the Crofoot transform as defined in [40]—as we will now show.

From Proposition 5.8, $M_{J_{{\overline{h}}}}$ will be an isometric isomorphism from ${\mathcal {K}}_{\theta }$ onto ${\text {ker}}T_{\overline{\theta _{{\overline{h}}}}}$ if and only if

$$\begin{aligned} 1 - \frac{|k|^2}{|1 - h \theta |^2} \in \overline{\theta H^2} + \theta H^2. \end{aligned}$$

(5.10)

Observe that

$$\begin{aligned} 1 - \frac{|k|^2}{|1 - h \theta |^2}&= \frac{(1 - h \theta )(1 - {\overline{h}} {\overline{\theta }}) - |k|^2}{(1 - h \theta )(1 - {\overline{h}} {\overline{\theta }})}\\&= \frac{h ({\overline{h}} - \theta ) + {\overline{h}} (h - {\overline{\theta }}) + 1 - h {\overline{h}} - |k|^2}{(1 - h \theta )(1 - {\overline{h}} {\overline{\theta }})}\\&= - \frac{h}{1 - h \theta } \theta - \frac{{\overline{h}}}{1 - {\overline{h}} {\overline{\theta }}} {\overline{\theta }} + \frac{1- |h|^2 - |k|^2}{(1 - h \theta )(1 - {\overline{h}} {\overline{\theta }})} \end{aligned}$$

and that

$$\begin{aligned} \frac{h}{1 - h \theta } \in H^{\infty } \subseteq H^2 \quad \text{ and } \quad \frac{{\overline{h}}}{1 - {\overline{h}} {\overline{\theta }}} \in \overline{H^{\infty }} \subseteq \overline{H^2}. \end{aligned}$$

Therefore, (5.10) is satisfied if and only if

$$\begin{aligned} \frac{1 - |h|^2 - |k|^2}{(1 - h \theta )(1 - {\overline{h}} {\overline{\theta }})} \in \overline{\theta H^2} + \theta H^2. \end{aligned}$$

(5.11)

This last condition is certainly satisfied when $h = {\overline{p}} \in {\mathbb {D}}$ is a constant function, and then choosing the constant k such that $|k|^2 = 1 - |h|^2$. In this case, $\theta _{{\overline{h}}} = \theta _{p}$ is an inner function (a Frostman shift of $\theta$ from (5.6)), ${\text {ker}}T_{\overline{\theta _{{\overline{h}}}}} = {\mathcal {K}}_{\theta _p}$ is a model space, and $M_{J_{{\overline{h}}}} = M_{J_p}$ is the Crofoot transform which maps ${\mathcal {K}}_{\theta }$ isometrically onto ${\mathcal {K}}_{\theta _p}$ (considered in [40]). Whether (5.11) can be satisfied for some $k \in {\mathbb {C}}\setminus \{0\}$ when h is not a constant function is an open question worthy of further investigation.

6 Truncated Toeplitz operators

In this section, we continue (and refine) our discussion of truncated Toeplitz operators begun in Sect. 2. From Theorem 2.28, with $\theta = \alpha , \eta = \gamma$, and $\varphi \in L^{\infty }$, and from Corollary 2.21, we have the following.

Theorem 6.1

If $\alpha$ and $\gamma$ are equivalent inner functions and $\alpha = \overline{a^{-1}} \gamma a$ with $a \in {\mathscr {G}}\!H^{\infty }$, then ${\mathcal {K}}_{\alpha } = a {\mathcal {K}}_{\gamma }$ and, for any $\varphi \in L^{\infty }$,

$$\begin{aligned} A_{{\overline{a}}}^{\alpha , \gamma } A_{\varphi }^{\alpha } A_{a}^{\gamma , \alpha } = A^{\gamma }_{\varphi |a|^2}. \end{aligned}$$

(6.2)

The relation in (6.2) defines a unitary equivalence between $A_{\varphi }^{\alpha }$ and $A_{\varphi |a|^2}^{\gamma }$ if and only if

$$\begin{aligned} 1 - |a|^2 \in \overline{\gamma H^2} + \gamma H^2, \end{aligned}$$

or, equivalently,

$$\begin{aligned} 1 - |a|^{-2} \in \overline{\alpha H^2} + \alpha H^2, \end{aligned}$$

and, in this case,

$$\begin{aligned} A_{\varphi }^{\alpha } = a A_{{\widetilde{\varphi }}}^{\gamma } a^{-1} I_{{\text {d}}}|_{{\mathcal {K}}_{\alpha }}. \end{aligned}$$

(6.3)

An example where (6.3) holds is given by the Crofoot transform acting on a truncated Toeplitz operator, as shown in [40] (see also Example 5.9).

Note that if we consider an operator on ${\mathcal {K}}_{\alpha }$ by replacing $A_{{{\overline{a}}}}^{\alpha , \gamma }$ in (6.2) by the inverse of $A^{\gamma , \alpha }_{a}$, i.e., multiplication by $a^{-1}$ on ${\mathcal {K}}_{\alpha }$, we do not, in general, obtain a truncated Toeplitz operator. This can be seen with the following example.

Example 6.4

For any $\lambda \in {\overline{{\mathbb {D}}}}$, and any finite Blaschke product $\theta$, define

$$\begin{aligned} k_{\lambda }^{\theta }(z) = \frac{1 - \overline{\theta (\lambda )} \theta (z)}{1 - {\overline{\lambda }} z} \quad \text{ and } \quad {\widetilde{k}}_{\lambda }^{\theta }(z) = \frac{\theta (z) - \theta (\lambda )}{z - \lambda }. \end{aligned}$$

Now let $\alpha$ be the simple degree two Blaschke product defined by

$$\begin{aligned} \alpha (z) = \frac{(z - \tfrac{1}{2})(z - \tfrac{1}{3})}{(1 - \tfrac{1}{2} z)(1 - \tfrac{1}{3} z)}, \end{aligned}$$

which can be factored as $\alpha = {\overline{a}}^{-1} z^2 a,$ where

$$\begin{aligned} a(z) = \frac{1}{(1 - \tfrac{1}{2} z)(1 - \tfrac{1}{3} z)} \in {\mathscr {G}}\!H^{\infty }, \end{aligned}$$

and let $\varphi (z) = \alpha (z)/z$. Then defining

$$\begin{aligned} A:= A_{a^{-1}}^{\alpha , z^2} A_{\alpha /z}^{\alpha } A_{a}^{z^2, \alpha } = a^{-1} A_{\alpha /z}^{\alpha } a I_{{\text {d}}}|_{{\mathcal {K}}_{z^2}}, \end{aligned}$$

which is an operator from ${\mathcal {K}}_{z^2}$ to itself, we see that for any $f \in {\mathcal {K}}_{z^2} = {\text {span}}\{1, z\}$,

$$\begin{aligned} A f = a(0) f(0) a^{-1} {\widetilde{k}}_{0}^{\alpha } = a(0) a^{-1} \cdot {\widetilde{k}}_{0}^{\alpha } \otimes k_{0}^{z^2} (f). \end{aligned}$$

(6.5)

Thus, A is a rank-one operator. If A were a truncated Toeplitz operator, it would have be a scalar multiple of one of the following types: $k_{\lambda }^{z^2} \otimes {\widetilde{k}}_{\lambda }^{z^2}$, ${\widetilde{k}}_{\lambda }^{z^2} \otimes k_{\lambda }^{z^2}$ for $\lambda \in {\mathbb {D}}$ or $k_{\zeta }^{z^2} \otimes k_{\zeta }^{z^2}$, where $\zeta \in {\mathbb {T}}$ [40]. Comparing this with the form of A in (6.5), we see that $a^{-1} {\widetilde{k}}_{0}^{\alpha }$ should be a constant multiple of ${\widetilde{k}}_{0}^{z^2}$, which it is not.

Theorem 6.1 allows us to reduce the study of truncated Toeplitz operators on arbitrary model spaces of dimension n to that of a truncated Toeplitz operator on ${\mathcal {K}}_{z^n}$, as in the following example.

Example 6.6

Let $\alpha$ be the degree two Blaschke product from Example 6.4 and consider the question of invertibility of the operator $A_{\varphi }^{\alpha }$, where

$$\begin{aligned} \varphi (z) = \frac{(z - \tfrac{1}{2})(z - \tfrac{1}{3}) (z^2 + 1)}{z^2}. \end{aligned}$$

Since $\alpha = {\overline{a}}^{-1} z^2 a$ with $a \in {\mathscr {G}}\!H^{\infty }$ as in Example 6.4, we can use Theorem 6.1 to conclude that $A_{\varphi }^{ \alpha }$ is equivalent to $A_{{\widetilde{\varphi }}}^{z^2}$ with

$$\begin{aligned} {\widetilde{\varphi }}(z) = 1 + \tfrac{5}{6} z + z^2 h_{+} \end{aligned}$$

for some $h_{+} \in H^{\infty }$. Moreover

$$\begin{aligned} A_{{\widetilde{\varphi }}}^{z^2} = A_{1 + \frac{5}{6} z}^{z^2} \end{aligned}$$

which is invertible [10, Thm. 4.1]. Thus, $A_{\varphi }^{\alpha }$ is invertible.

Also note that (2.30) provides an equivalence between the inverses of the two operators $A_{\varphi }^{\alpha }$ and $A_{{\widetilde{\varphi }}}^{\gamma }$, if they exist. We have the following result which can be obtained from [10, Thm. 2.4].

Proposition 6.7

Let $\theta$ be an inner function and let $\varphi \in L^{\infty }$ be such that

$$\begin{aligned} G = \begin{bmatrix} {\overline{\theta }} &{}\quad 0\\ \varphi &{}\quad \theta \end{bmatrix} \end{aligned}$$

admits a Wiener–Hopf factorization of the form $G = G_{-} G_{+}$, with ${\overline{G}}_{-}^{\,\pm 1}, G_{+}^{\pm 1} \in (H^{\infty })_{2 \times 2}$ [10, 37]. Moreover, denote $G_{+}^{-1} = [g_{ij}^{+}]$ and $G_{-}^{-1} = [g_{ij}^{-}]$, $i, j = 1, 2.$ Then, for all $f \in {\mathcal {K}}_{\theta }$,

$$\begin{aligned} (A_{\varphi }^{\theta })^{-1} f = P_{\theta }(g_{11}^{+} P_{+} g_{12}^{-} + g_{12}^{+} P_{+} g_{22}^{-}) f. \end{aligned}$$

When $\theta = z^n$, necessary and sufficient conditions for $A_{\varphi }^{\theta }$ to be invertible were obtained in [10] and the factorization of G itself can be similarly determined in explicit form by solving the relatively simple Riemann–Hilbert problem $G G_{+}^{-1} = G_{-}$ (G is a $2 \times 2$ triangular matrix of polynomials with diagonal entries ${\overline{z}}^n$ and $z^n$). Proposition 6.7 provides an explicit and simple expression for the inverse of $A_{{\widetilde{\varphi }}}^{z^n}$ and from (2.30) we can then obtain an expression for the inverse of $A_{\varphi }^{\alpha }$ in a simple way.

As we have seen in Example 5.3, for two finite Blaschke products $\alpha$ and $\gamma$ of equal degree, there is an $a \in {\mathscr {G}} H^{\infty }$ such that $a {\mathcal {K}}_{\gamma } = {\mathcal {K}}_{\alpha }$. This allows us to prove the following.

Proposition 6.8

Let $\alpha$ and $\gamma$ be two finite Blaschke products of equal degree. Then two truncated Toeplitz operators $A^{\alpha }_{\varphi }$ and $A^{\gamma }_{\psi }$ are equivalent if and only if they have the same rank.

Proof

This result will follow from the following linear algebra result. Two $n \times n$ matrices A and B are equivalent (i.e., there are $n \times n$ invertible matrices E and F with $A = E B F$) if and only if they have the same rank. Indeed, if A is equivalent to B then clearly A and B have the same rank. Conversely, if A and B have the same rank $r \leqslant n$ then by the singular value decomposition there are $n \times n$ unitary matrices U, V, W, Q such that $A = U \Sigma _{A} V$ and $B = W \Sigma _{B} Q,$ where $\Sigma _{A}, \Sigma _{B}$ are the $n \times n$ diagonal matrices

$$\begin{aligned}{} & {} \Sigma _{A} = {\text {diag}}(\sigma _1, \sigma _2, \ldots , \sigma _r, 0, 0, \ldots 0), \\{} & {} \Sigma _{B} = {\text {diag}}(\lambda _1, \lambda _2, \ldots , \lambda _r, 0, 0, \ldots , 0). \end{aligned}$$

Of course, $\sigma _j$ and $\lambda _j$ are nonzero for all $1 \leqslant j \leqslant r$. If

$$\begin{aligned} D = {\text {diag}}\left( \frac{\sigma _1}{\lambda _1}, \frac{\sigma _2}{\lambda _2}, \ldots , \frac{\sigma _r}{\lambda _r}, 1, 1, \ldots , 1\right) , \end{aligned}$$

then D is invertible and

$$\begin{aligned} A&= U \Sigma _{A} V\\&= U D \Sigma _{B} V\\&= U D W^{*} B Q^{*} V\\&= (U D W^{*}) B (Q^{*} V)\\&= E B F \end{aligned}$$

and so A is equivalent to B. $\square$

The proof of Proposition 6.8 says that any matrix is equivalent to a Toeplitz matrix of equal rank. In fact, one can say more. A result from [13, Thm. 6.2] says that any square matrix is similar to some truncated Toeplitz operator on some finite dimensional model space. One can only go so far with these type of results since small matrices are similar to Toeplitz matrices while large ones are generally not [33].

We end with an application to the essential spectrum of two equivalent truncated Toeplitz operators. For a bounded operator T on a Hilbert space ${\mathcal {H}}$, recall that the essential spectrum of T, denoted by $\sigma _{e}(T)$, is the set of all $\lambda \in {\mathbb {C}}$ such that the operator $T - \lambda I$ is not Fredholm.

For an inner function $\alpha$, let

$$\begin{aligned} \Sigma _{{\mathbb {T}}}(\alpha ):= \left\{ \xi \in {\mathbb {T}}: \varliminf _{z \rightarrow \xi } |\alpha (z)| = 0\right\} \end{aligned}$$

denote the boundary spectrum of the inner function $\alpha$. One can show that $\Sigma _{{\mathbb {T}}}(\alpha )$ consists of the accumulation points of the zeros of $\alpha$ along with the support of the associated singular measure for $\alpha$ in its Riesz factorization [24, p. 152]. Another known result [24, p. 204] is that

$$\begin{aligned} \Sigma _{{\mathbb {T}}}(\alpha ) = \sigma _{e}(A^{\alpha }_{z}). \end{aligned}$$

(6.9)

Let ${\mathscr {R}}$ denote the set of rational functions whose poles are not in ${\mathbb {T}}$. Also define

$$\begin{aligned} {\mathscr {R}}_{+}:= P_{+} {\mathscr {R}} = {\mathscr {R}} \cap H^{\infty } \quad \text{ and } \quad {\mathscr {R}}_{-}:= P_{-} {\mathscr {R}} = {\mathscr {R}} \cap \overline{z H^{\infty }}. \end{aligned}$$

As used before, $P_{+}$ is the Riesz projection from $L^2$ onto $H^2$ and $P_{-} = I - P_{+}$. If $\gamma$ is inner and also satisfies $\alpha = {\overline{a}}^{-1} \gamma a$ for some $a \in {\mathscr {G}}\!H^{\infty }$, then by Theorem 2.28, the operators $A^{\alpha }_{R - \lambda }$ and $A^{\gamma }_{|a|^2 (R - \lambda )}$ are equivalent for all $\lambda \in {\mathbb {C}}$. Thus, $A^{\alpha }_{R - \lambda }$ is Fredholm if and only if the operator $A^{\gamma }_{|a|^2 (R - \lambda )}$ is Fredholm. Using the notation $P_{\gamma H^2}$ for the orthogonal projection of $L^2$ onto $\gamma H^2$, we see that

$$\begin{aligned} A^{\gamma }_{|a|^2 (R - \lambda )}&= P_{\gamma } |a|^2(R - \lambda ) P_{\gamma }|_{{\mathcal {K}}_{\gamma }}\\&= \Big (P_{\gamma } (R - \lambda ) P_{\gamma } {\overline{a}} P_{\alpha } a P_{\gamma } + P_{\gamma } (R - \lambda ) P_{-} ({\overline{a}} P_{\alpha } a P_{\gamma })\\&\quad + P_{\gamma } (R - \lambda ) P_{\gamma H^2} ({\overline{a}} P_{\alpha } a P_{\gamma })\Big )|_{{\mathcal {K}}_{\gamma }}\\&= A_{R - \lambda }^{\gamma } A_{{\overline{a}}}^{\alpha , \gamma } A_{a}^{\gamma , \alpha } + P_{\gamma } (R - \lambda ) P_{-} ({\overline{a}} P_{\alpha } a P_{\gamma })|_{{\mathcal {K}}_{\gamma }}\\&\quad + P_{\gamma } (R - \lambda ) P_{\gamma H^2} ({\overline{\alpha }} P_{\alpha } \alpha P_{\gamma })|_{{\mathcal {K}}_{\gamma }}, \end{aligned}$$

where the last two operators above are finite rank (see Lemma 6.11 below). Since the operators $A_{{\overline{a}}}^{\alpha , \gamma }$ and $A_{a}^{\gamma , \alpha }$ are invertible, we obtain the following.

Theorem 6.10

If $\alpha$ and $\gamma$ are inner functions with $\alpha \sim \gamma$ and $R \in {\mathscr {R}}$, then $\sigma _{e}(A^{\alpha }_{R}) = \sigma _{e}(A^{\gamma }_{R})$.

As mentioned above, the needed ingredient to complete the proof of Theorem 6.10 is the following lemma.

Lemma 6.11

If $R \in {\mathscr {R}}$, then $P_{\gamma } R P_{-}$ and $P_{\gamma } R P_{\gamma H^2}$ are finite rank operators.

Proof

We have $R = R_{+} + R_{-}$ with $R_{\pm } \in {\mathscr {R}}_{\pm }$. Then

$$\begin{aligned} P_{\gamma } R P_{-} = P_{\gamma } (R_{-} + R_{+}) P_{-} = P_{\gamma } (P_{+} R_{+} P_{-}) \end{aligned}$$

and

$$\begin{aligned} P_{\gamma } R P_{\gamma H^2}&= P_{\gamma } (R_{-} + R_{+}) \gamma P_{+} {\overline{\gamma }} P_{+}\\&= P_{\gamma } R_{-} \gamma P_{+} {\overline{\gamma }} P_{+}\\&= P_{\gamma } \gamma (P_{-} R_{-} P_{+}) {\overline{\gamma }} P_{+}. \end{aligned}$$

By Kronecker’s theorem on finite rank Hankel operators [39], it follows that $P_{\gamma } R_{+} P_{-}$ and $P_{\gamma } R_{-} P_{\gamma H^2}$ are finite rank operators. $\square$

Theorem 6.10 along with (6.9) gives another proof of an observation of Crofoot [14, Thm. 14]: A necessary condition for $\alpha$ to be equivalent to $\gamma$ is that $\Sigma _{{\mathbb {T}}}(\alpha ) = \Sigma _{{\mathbb {T}}}(\gamma )$.

7 Dual truncated Toeplitz operators

This section formulates a version of Theorem 2.12 when the generalized Toeplitz operator becomes a dual truncated Toeplitz operator. If $\alpha$ is inner, recall from our discussion at the end of Sect. 2 that $P_{\alpha }$ denotes the orthogonal projection of $L^2$ onto the model space ${\mathcal {K}}_{\alpha }$. Then $Q_{\alpha }:= I - P_{\alpha }$ is the orthogonal projection of $L^2$ onto

$$\begin{aligned} {\mathcal {K}}_{\alpha }^{\perp } = \alpha H^{2} \oplus (H^{2})^{\perp }. \end{aligned}$$

For $\varphi \in L^{\infty }$, define

$$\begin{aligned} D_{\varphi }^{\gamma , \alpha }: {\mathcal {K}}_{\gamma }^{\perp } \rightarrow {\mathcal {K}}_{\alpha }^{\perp }, \quad D_{\varphi }^{\gamma , \alpha }f:= Q_{\alpha } (\varphi f). \end{aligned}$$

When $\gamma = \alpha$ we write $D^{\gamma , \gamma }_{\varphi } = D_{\varphi }^{\gamma }$. The operator $D_{\varphi }^{\gamma }$ is a dual truncated Toeplitz operator on ${\mathcal {K}}_{\gamma }$ with symbol $\varphi$, while $D_{\varphi }^{\gamma , \alpha }$ is an asymmetric dual truncated Toeplitz operator from ${\mathcal {K}}_{\gamma }$ to ${\mathcal {K}}_{\alpha }$ with symbol $\varphi$. These operators have received considerable attention recently [8, 16,17,18]. Unlike the criterion in (2.26) and (2.27) for an asymmetric truncated Toeplitz operator to be the zero operator, we have the following (see [16, Prop. 2.1])

$$\begin{aligned} D^{\gamma , \alpha }_{\varphi } \equiv 0 \iff \varphi = 0. \end{aligned}$$

(7.1)

Thus, the symbol of a dual truncated Toeplitz operator is unique.

Some results concerning the equivalence of dual truncated Toeplitz operators are known when $\varphi (z) = z$ and thus $D_{z}^{\alpha }$ is the dual of the compressed shift [7]. For example, $D_{z}^{\alpha }$ is unitarily equivalent to $D_{z}^{\gamma }$ if and only if $|\alpha (0)| = |\gamma (0)|$. When $\alpha (0) \not = 0$ and $\gamma (0) \not = 0$, then $D_{z}^{\alpha }$ is similar to $D_{z}^{\gamma }$ and they are both similar to the bilateral shift on $L^2$ (i.e., the operator $f(\xi ) \mapsto \xi f(\xi )$). When $\alpha (0) = 0$, $D_{z}^{\alpha }$ is unitarily equivalent to $S \oplus S^{*}$ on $H^2 \oplus H^2$, where S is the unilateral shift. From here, one can characterize the invariant subspaces of $D_{z}^{\alpha }$ [43].

A version of Theorem 2.12 for dual truncated Toeplitz operators is the following.

Theorem 7.2

Let $\theta$, $\eta , \alpha , \gamma$ be inner functions, $\theta \sim \eta$ and $\alpha \sim \gamma$ with $\theta = a_1 \eta \overline{a_{1}^{-1}}$ and $\alpha = a_2 \gamma \overline{a_{2}^{-1}}$, where $a_{1}, a_{2} \in {\mathscr {G}}\! \overline{H^{\infty }}$. Then for any $\varphi \in L^{\infty }$, the operator $D_{\varphi }^{\theta , \alpha }$ is equivalent to $D_{{\widetilde{\varphi }}}^{\eta , \gamma }$, where ${\widetilde{\varphi }} = {\overline{a}}_2 \varphi a_1$. Moreover, $D_{{\overline{a}}_{2}^{-1}}^{\gamma , \alpha }$ and $D_{a_{1}^{-1}}^{\theta , \eta }$ are bounded invertible operators such that

$$\begin{aligned} D_{\varphi }^{\theta , \alpha } = D_{{\overline{a}}_{2}^{-1}}^{\gamma , \alpha } D_{{\widetilde{\varphi }}}^{\eta , \gamma } D_{a_{1}^{-1}}^{\theta , \eta }. \end{aligned}$$

Corollary 7.3

For inner functions $\gamma$, $\alpha$ and $a \in {\mathscr {G}} \overline{H^{\infty }}$ with $\alpha = a \gamma {\overline{a}}^{-1}$, let $\varphi \in L^{\infty }$ and ${\widetilde{\varphi }} = |a|^2 \varphi$. Then $D^{\alpha }_{\varphi } = (D^{\alpha , \gamma }_{a^{-1}})^{*} D^{\gamma }_{{\widetilde{\varphi }}} D_{a^{-1}}^{\alpha , \gamma }.$

One can wonder whether the equivalence in the previous result can be an isometric isomorphism, that is to say, $D_{a}^{\gamma , \alpha } = D_{{\overline{a}}^{-1}}^{\gamma , \alpha }$? Unlike the case with truncated Toeplitz operators discussed earlier (Theorem 6.1), similarity in this case can only happen in a trivial way. Indeed, $D_{a}^{\gamma , \alpha } = D_{{\overline{a}}^{-1}}^{\gamma , \alpha }$ implies that $D_{a -{\overline{a}}^{-1}}^{\gamma , \alpha } \equiv 0$ and so, by (7.1), $a = {\overline{a}}^{-1}$. Since $H^{\infty } \cap \overline{H^{\infty }} = {\mathbb {C}}$, this would imply that $a = {\overline{a}}^{-1}$ and is a unimodular constant function.

It follows from Corollary 7.3 that

$$\begin{aligned} {\text {ker}}D^{\theta }_{\varphi } = {\overline{a}} {\text {ker}}D_{{\widetilde{\varphi }}}^{\eta }. \end{aligned}$$

(7.4)

As an application of this result we study the kernel of a dual truncated Toeplitz operator whose symbol $\varphi$ does not belong to ${\mathscr {G}}\!L^{\infty }$ and thus whose study cannot be reduced to that of a truncated Toeplitz operator with symbol $\varphi ^{-1}$ [8, §6].

We have that

$$\begin{aligned} {\text {ker}}D^{\theta }_{\varphi } = \{f_1 \in {\mathcal {K}}_{\theta }^{\perp }: \varphi f_1 = f_{2+} \in {\mathcal {K}}_{\theta }\}. \end{aligned}$$

(7.5)

Noting that $f_1 \in {\mathcal {K}}_{\theta }^{\perp }$ if and only if $f_{1} = f_{1-} + \theta f_{1+}$, where $f_{1-} \in (H^2)^{\perp }$ and $f_{1+} \in H^2$; and $f_{2+} \in {\mathcal {K}}_{\theta }$ if and only if $f_{2+} \in H^2$ and ${\overline{\theta }} f_{2+} = f_{2-} \in (H^2)^{\perp }$, the equation defining ${\text {ker}}D_{\varphi }^{\theta }$ is equivalent to the matricial Riemann–Hilbert problem

$$\begin{aligned} \begin{bmatrix} {\overline{\theta }} &{}\quad 0\\ -1 &{}\quad \varphi \theta \end{bmatrix} \begin{bmatrix} f_{2+}\\ f_{1+}\end{bmatrix} + \begin{bmatrix} -1 &{}\quad 0\\ 0 &{}\quad \varphi \end{bmatrix} \begin{bmatrix} f_{2-}\\ f_{1-}\end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \end{aligned}$$

Now let $\theta$ be a finite Blaschke product of degree n and, for any inner function $\alpha$, define $\varphi (z) = \alpha (z) (z - 1)$. From Example 5.3 we see that

$$\begin{aligned} \theta = \overline{B_{+}^{-1}} z^n B_{+} \quad \text{ with } B_{+} \in {\mathscr {G}}\!H^{\infty }. \end{aligned}$$

(7.6)

Therefore, by Corollary 7.3,

$$\begin{aligned} D_{\varphi }^{\theta } = D_{B_{+}}^{z^n, \theta } D_{{\widetilde{\varphi }}}^{z^n} D_{\overline{B_{+}}}^{\theta , z^n}, \quad \text{ where } \quad {\widetilde{\varphi }} = \overline{B_{+}^{-1}} \varphi B_{+}^{-1}, \end{aligned}$$

and, by (7.4),

$$\begin{aligned} {\text {ker}}D_{\varphi }^{\theta } = D^{z^n, \theta }_{\overline{B_{+}^{-1}}} {\text {ker}}D_{{\widetilde{\varphi }}}^{z^n} = \overline{B_{+}^{-1}} {\text {ker}}D_{{\widetilde{\varphi }}}^{z^n}. \end{aligned}$$

Thus we have reduced the problem of determining ${\text {ker}}D^{\theta }_{\varphi }$ to that of determining ${\text {ker}}D_{{\widetilde{\varphi }}}^{z^n}$. For this problem, we use (7.5) with $\theta$ and $\varphi$ replaced by $z^n$ and ${\widetilde{\varphi }}$ respectively to obtain

$$\begin{aligned} {\overline{z}}^n f_{2+} = f_{2-} \end{aligned}$$

(7.7)

and

$$\begin{aligned} f_{2+} - B_{+}^{-1} \alpha (z - 1) \overline{B_{+}^{-1}} z^n f_{1+} = B_{+}^{-1} \alpha (z - 1) \overline{B_{+}^{-1}} f_{1-}. \end{aligned}$$

(7.8)

From (7.7) we have that $f_{2+} = p_{n - 1}$, where $p_{n - 1} \in {\mathscr {P}}_{n - 1}$ (the polynomials of degree at most $n - 1$). Substituting this into the equation in (7.8) we see that

$$\begin{aligned} p_{n - 1} - B_{+}^{-1} \alpha (z - 1) \overline{B_{+}^{-1}} z^n f_{1+} = B_{+}^{-1} \alpha (z - 1) \overline{B_{+}^{-1}} f_{1-} \end{aligned}$$

if and only if

$$\begin{aligned} B_{+}^{-1} (z - 1) f_{1+} = {\overline{\alpha }} {\overline{z}}^{n} p_{n - 1} \overline{B_{+}} - {\overline{\theta }} \overline{B_{+}^{-1}} (z - 1) f_{1-}. \end{aligned}$$

Moreover, since the left hand side of the last equality belongs to $H^2$ while the right hand side belongs to $\overline{H^2}$, both sides are equal to a constant. Due to the factor of $z - 1$ on the left hand side, this constant must be zero. This means that $f_{1+} = 0$ and

$$\begin{aligned} {\overline{\alpha }} {\overline{z}}^n p_{n - 1} \overline{B_{+} }= {\overline{\theta }} \overline{B_{+}^{-1}} (z - 1) f_{1-}. \end{aligned}$$

(7.9)

It follows from (7.9) that

$$\begin{aligned} \frac{p_{n - 1}}{z - 1} = {\overline{\theta }} \alpha z^n \overline{B_{+}^{-2}} f_{1-} \in L^2 \end{aligned}$$

and so $p_{n - 1}(1) = 0$. Therefore, if $n = 1$ we have that $p_{n - 1} \equiv 0$ and thus ${\text {ker}}D_{\varphi }^{\theta } = \{0\}$. When $n > 1$, we have

$$\begin{aligned} {\overline{\alpha }} {\overline{z}}^n {\overline{B}}_{+}^{2} p_{n - 2} = {\overline{\theta }} f_{1-} \quad \text{ with } p_{n - 2} \in {\mathscr {P}}_{n - 2}. \end{aligned}$$

(7.10)

Let $\gamma = {\text {gcd}}(\theta , z \alpha )$ (which is necessarily a finite Blaschke product). Then, from (7.10),

$$\begin{aligned} \frac{z^2 \alpha }{\gamma } B_{+}^{2} z^{n - 2} \overline{p_{n - 2}} = \frac{\theta }{\gamma } \overline{f_{1-}}. \end{aligned}$$

From the fact that $z^{n - 2} \overline{p_{n - 2}} = {\widetilde{p}}_{n - 2} \in {\mathscr {P}}_{n - 2}$, we see that

$$\begin{aligned} \frac{z \alpha }{\gamma } B_{+}^{2} {\widetilde{p}}_{n - 2} = \frac{\theta }{\gamma } {\overline{z}} \overline{f_{1-}} \quad \hbox { with}\ {\overline{z}} \overline{f_{1-}} \in H^2. \end{aligned}$$

and thus,

$$\begin{aligned} {\widetilde{p}}_{n - 2} \in \frac{\theta }{\gamma } H^2. \end{aligned}$$

Let $k = {\text {deg}}(\theta /\gamma )$. When $n - 2 < k$, we have ${\widetilde{p}}_{n - 2} = 0$ and so ${\text {ker}}D_{\varphi }^{\theta } = \{0\}$. When $n - 2 \geqslant k$ we have ${\widetilde{p}}_{n - 2} = p_{\theta /\gamma } {\widetilde{p}}_{n - 2 - k},$ where $p_{\theta /\gamma }$ is the numerator of $\theta /\gamma$ and ${\widetilde{p}}_{n - 2 - k} \in {\mathscr {P}}_{n - 2 - k}$. Hence,

$$\begin{aligned} f_{1-} = \theta \overline{p_{\theta /\gamma }} {\overline{\alpha }} {\overline{z}}^{2} \overline{B_{+}^{2}} \overline{{\widetilde{p}}_{n - 2 - k}}. \end{aligned}$$

Conversely, if $f_{1} = f_{1-}$ (note that $f_{1+} = 0$), then $f_{1} \in {\text {ker}}D_{{\widetilde{\varphi }}}^{z^n}$ and so ${\text {dim}} {\text {ker}}D_{\varphi }^{\theta } = n - 1 - k$.

Summarizing these results we obtain the following.

Proposition 7.11

Let $\theta$ be a Blaschke product of degree n satisfying (7.6) and, for any inner function $\alpha$, let $\varphi (z) = \alpha (z) (z - 1).$ Furthermore, let

$$\begin{aligned} \gamma = {\text {gcd}}(\theta , z \alpha ), \quad k = {\text {deg}} \frac{\theta }{\gamma }, \end{aligned}$$

and

$$\begin{aligned} d(z) = \prod _{j = 1}^{k} (1 - \overline{w_{j}} z), \end{aligned}$$

where $w_1, \ldots , w_k$ are the zeros of $\theta /\gamma$ (counting multiplicity). Then

• (a) if $n \leqslant k + 1$, ${\text {ker}}D_{\varphi }^{\theta } = \{0\}$.

• (b) if $n > k + 1$,

  $$\begin{aligned} {\text {ker}}D_{\varphi }^{\theta } = \overline{B_{+}} \cdot {\overline{d}} \cdot \overline{\left( \frac{\alpha z}{\gamma }\right) } \cdot {\overline{z}} \cdot \overline{{\mathscr {P}}_{n - k - 2}} \subseteq (H^2)^{\perp } \end{aligned}$$

  and ${\text {dim}} {\text {ker}}D_{\varphi }^{\theta } = n - 1 - k$.