Abstract
Conjugations in space \(L^2\) of the unit circle commuting with multiplication by z or intertwining multiplications by z and \({{\bar{z}}}\) are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.
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1 Introduction
Let \({\mathcal {H}}\) be a complex Hilbert space and denote by \(B({{\mathcal {H}}})\) the algebra of all bounded linear operators on \({\mathcal {H}}\). A conjugationC in \({\mathcal {H}}\) is an antilinear isometric involution, i.e., \(C^2=id_{{\mathcal {H}}}\) and
Conjugations have recently been intensively studied and the roots of this subject comes from physics. An operator \(A\in B({{\mathcal {H}}})\) is called C–symmetric if \(CAC=A^*\) (or equivalently \(AC=CA^*\)). A strong motivation to study conjugations comes from the study of complex symmetric operators, i.e., those operators that are C–symmetric with respect to some conjugation C. For references see for instance [2, 3, 7,8,9,10]. Hence obtaining the full description of conjugations with certain properties is of great interest.
Let \({\mathbb {T}}\) denote the unit circle, and let m be the normalized Lebesgue measure on \({\mathbb {T}}\). Consider the spaces \(L^2=L^2(\mathbb {T,}m)\), \(L^{\infty }=L^\infty ({\mathbb {T}},m)\), the classical Hardy space \(H^2\) on the unit disc \({\mathbb {D}}\) identified with a subspace of \(L^2\), and the Hardy space \(H^\infty \) of all analytic and bounded functions in \({\mathbb {D}}\) identified with a subspace of \(L^\infty \). Denote by \(M_\varphi \) the operator defined on \(L^2\) of multiplication by a function \(\varphi \in L^\infty \).
The most natural conjugation in \(L^2\) is J defined by \(Jf={{\bar{f}}}\), for \(f\in L^2\). This conjugation has two natural properties: the operator \(M_z\) is J–symmetric, i.e., \(M_z J=J M_{{{\bar{z}}}}\), and J maps an analytic function into a co-analytic one, i.e., \(J H^2=\overline{H^2}\).
Another natural conjugation in \(L^2\) is \(J^{\star }f=f^{\#}\) with \(f^{\#}(z)\overset{df}{=}\overline{ f({{\bar{z}}})}.\) The conjugation \(J^{\star }\) has a different behaviour: it commutes with multiplication by z (\(M_z J^\star =J^\star M_{ z}\)) and leaves analytic functions invariant, \(J^\star H^2\subset {H^2}\). The map \(J^{\star }\) appears for example in connection with Hankel operators (see [11, pp. 146–147]). Its connection with model spaces was studied in [4] (Lemma 4.4, see also [1, p. 37]). Hence a natural question is to characterize conjugations with respect to these properties. The first step was done in [2] where all conjugations in \(L^2\) with respect to which the operator \(M_z\) is C–symmetric were characterized, see Theorem 2.2. In Sect. 2 we give a characterization of all conjugations which commute with \(M_z\), Theorem 2.4. In Sect. 3, using the above characterizations we show that there are no conjugations in \(L^2\) leaving \(H^2\) invariant, with respect to which the operator \(M_z\) is C–symmetric. We also show that \(J^{\star }\) is the only conjugation commuting with \(M_z\) and leaving \(H^2\) invariant.
Beurling’s theorem makes subspaces of \(H^2 \) of the type \(\theta H^2\) (\(\theta \) inner function, i.e., \(\theta \in H^\infty \), \(|\theta |=1\) a.e. on \({\mathbb {T}}\)) exceptionally interesting, as the only invariant subspaces for the unilateral shift S, \(Sf(z)=zf(z)\) for \(f\in H^2\). On the other hand, model spaces (subspaces of the type \(K_\theta =H^2\ominus \theta H^2\)), which are invariant for the adjoint of the unilateral shift, are important in model theory, [12].
In [2] all conjugations C with respect to which the operator \(M_z\) is C–symmetric and mapping a model space \(K_\alpha \) into another model space \(K_\theta \) were characterized, with the assumption that \(\alpha \) divides \(\theta \) (\(\alpha \leqslant \theta \)). Recall that for \(\alpha \) and \(\theta \) inner, \(\alpha \leqslant \theta \) means that \(\theta /\alpha \) is also an inner function. In what follows we will show that the result holds without the assumption \(\alpha \leqslant \theta \).
In Sect. 4 conjugations commuting with \(M_z\) and preserving model spaces are described. Section 5 is devoted to conjugations between S–invariant subspaces (i.e., subspaces of the form \(\theta H^2\) with \(\theta \) an inner function). In the last section we deal with conjugations commuting with the truncated shift \(A^\theta _z\) (\(A^\theta _z=P_{\theta }M_{z|K_{\theta }}\) where \(P_{\theta }\) is the orthogonal projection from \(L^2\) onto \(K_{\theta }\)) or conjugations such that \(A^\theta _z\) is C–symmetric with respect to them.
2 \(M_z\) and \(M_z\)–commuting conjugations in \(L^2\)
Denote by J the conjugation in \(L^2\) defined as \(Jf={{\bar{f}}}\), for \(f\in L^2\). This conjugation has the following obvious properties:
Proposition 2.1
-
(1)
\(M_z J=J M_{{{\bar{z}}}}\);
-
(2)
\(M_\varphi J=JM_{{{\bar{\varphi }}}}\) for all \(\varphi \in L^\infty \);
-
(3)
\(J H^2=\overline{H^2}\).
Let us consider all conjugations C in \(L^2\) satisfying the condition
Such conjugations were studied in [2] and are called \(M_z\)–conjugations. The following theorem characterizes all \(M_z\)–conjugations in \(L^2\).
Theorem 2.2
[2] Let C be a conjugation in \(L^2\). Then the following are equivalent:
-
(1)
\(M_z C=C M_{{{\bar{z}}}}\),
-
(2)
\(M_\varphi C=C M_{{{\bar{\varphi }}}}\) for all \(\varphi \in L^\infty \),
-
(3)
there is \(\psi \in L^\infty \), with \(|\psi |=1\), such that \(C=M_\psi J\),
-
(4)
there is \(\psi ^\prime \in L^\infty \), with \(|\psi ^\prime |=1\), such that \(C=JM_{\psi ^\prime }\).
Another natural conjugation in \(L^2\) is defined as
The basic properties of \(J^{\star }\) are the following:
Proposition 2.3
-
(1)
\(M_z J^{\star }=J^{\star } M_{ z}\);
-
(2)
\(M_{{{\bar{z}}}} J^{\star }=J^{\star } M_{{{\bar{z}}}}\);
-
(3)
\(M_\varphi J^{\star }=J^{\star }M_{\varphi ^{\#}}\) for all \(\varphi \in L^\infty \);
-
(4)
\(J^\star H^2=H^2\).
In the context of the Proposition 2.3 it seems natural to consider all conjugations C in \(L^2\) commuting with \(M_z\), i.e.,
Such conjugations will be called \(M_z\)–commuting. In what follows we will often deal with functions \(f\in L^2\) such that \(f(z)=f({{\bar{z}}})\) a.e. on \({\mathbb {T}}\), which will be called symmetric. Observe that if f is symmetric and \(f\in H^2\), then we also have \(f\in \overline{H^2}\) and so it is a constant function.
Theorem 2.4
Let C be a conjugation in \(L^2\). Then the following are equivalent:
-
(1)
\( M_z C=C M_{z}\) ,
-
(2)
\(M_\varphi C=C M_{\varphi ^{\#}}\) for all \(\varphi \in L^\infty \),
-
(3)
there is a symmetric unimodular function \(\psi \in L^\infty \) such that \(C=M_\psi J^{\star }\),
-
(4)
there is a symmetric unimodular function \(\psi ^\prime \in L^\infty \) such that \(C=J^{\star }\,M_{\psi ^\prime }\).
Proof
We will show that \((1)\!\Rightarrow \!(3)\). The other implications are straightforward. Assume that \(M_{ z}C =CM_z\). Then \(M_z CJ^{\star }=CM_{ z}J^{\star }=CJ^{\star }M_z\). It follows that the linear operator \(CJ^{\star }\) commutes with \(M_z\). By [13, Theorem 3.2], there is \(\psi \in L^\infty \) such that \(CJ^{\star }=M_{\psi }\). Hence \(C=M_{\psi }J^{\star }=J^{\star }M_{\psi ^{\#}}\).
By (1.1) for any \(f,g\in L^2\) we have
Hence \(|\psi |=1\) a.e. on \({\mathbb {T}}\). On the other hand, since \(C^2=I_{L^2}\), for all \(f\in L^2\) we have
which implies that \(\psi \psi ^{\#}=1\) a.e. on \({\mathbb {T}}\). Therefore \(\psi \) is symmetric, i.e., \(\psi (z)=\psi ({{\bar{z}}})\) a.e. on \({\mathbb {T}}\). \(\square \)
3 Conjugations preserving \(H^2\)
In the previous section all \(M_z\)–conjugations C in \(L^2\), i.e., such that
or \(M_z\)–commuting conjugations, i.e., such that
were characterized. Let us now consider the question which of them preserve \(H^2\). Clearly, if C is a conjugation in \(L^2\) and \(C(H^2)\subset H^2\), then \(C(H^2)= H^2\). Since \(J^\star \) preserves \(H^2\), it can be considered as a conjugation in \(H^2\). The following result shows that \(J^\star \) is in that sense unique.
Corollary 3.1
Let C be an \(M_z\)–commuting conjugation in \(L^2\). If \(C(H^2)\subset H^2\), then \(C=\lambda J^{\star }\) for some \(\lambda \in {\mathbb {T}}\).
Proof
By Theorem 2.4 we have that \(C=M_{\psi }J^{\star }\) for some \(\psi \in L^{\infty }\) with \(|\psi |=1\) and \(\psi (z)=\psi ({\overline{z}})\) a.e. on \({\mathbb {T}}\). Since C preserves \(H^2\), we have
Thus \(\psi \) is analytic. Since it is symmetric, it is also co-analytic. Hence \(\psi \) must be a constant function, so \(\psi =\lambda \in {\mathbb {C}}\) and \(|\lambda |=|\psi |=1\). \(\square \)
Corollary 3.2
There are no \(M_z\)–conjugations in \(L^2\) which preserve \(H^2\).
Proof
If C is an \(M_z\)–conjugation in \(L^2\), then by Theorem 2.2 it follows that \(C=M_{\psi }J\) for some \(\psi \in L^{\infty }\) with \(|\psi |=1\). As in the proof of Corollary 3.1 the assumption \(C(H^2)\subset H^2\) implies that \(\psi \in H^{\infty }\), which in turn means that \(\psi \) is an inner function. Moreover, for \(n=0,1,2,\dots \) we have
So \(\psi =0\) which is a contradiction. \(\square \)
The following example shows that not all conjugations in \(L^2\) satisfy either (2.1) or (2.3).
Example 3.3
There is a set of naturally defined conjugations. For \(k,l\in {\mathbb {Z}}\), \(k<l\), define \(C_{k,l}\ :\ L^2\rightarrow L^2\) by
where \(\{z^n\}\) is the standard basis in \(L^2\). Then (2.1) and (2.3) are not satisfied since
and
Note that, on the other hand, \(C_{k,l}\) preserves \(H^2\) whenever \(k\geqslant 0\) or \(l<0\).
4 Conjugations preserving model spaces
There is another class of conjugations in \(L^2\) which appear naturally in connection with model spaces. For a nonconstant inner function \(\theta \), denote by \(K_\theta \) the so called model space of the form \(H^2\ominus \theta H^2\). The conjugation \(C_\theta \) defined in \(L^2\) by
has the important property that it preserves the model space \(K_\theta \), i.e., \(C_\theta K_\theta =K_\theta \). Thus \(C_\theta \) can be considered as a conjugation in \(K_\theta \). Such conjugations are important in connection with truncated Toeplitz operators (see for instance [6]). Here we present several simple properties of such conjugations, which we will use later.
Proposition 4.1
Let \(\alpha ,\beta ,\gamma \) be nonconstant inner functions. Then
-
(1)
\(C_\beta C_\alpha =M_{\beta {{\bar{\alpha }}}}\),
-
(2)
\(M_{\gamma } C_\alpha M_{{{\bar{\gamma }}}}\) is a conjugation in \(L^2\),
-
(3)
\(C_\beta \, M_\gamma = M_{{{\bar{\gamma }}}}\,C_\beta \).
Now we will consider relations between \(M_z\)–conjugations and model spaces. The theorem below was proved in [2, Theorem 4.2] with the additional assumption that \(\alpha \leqslant \theta \). As we prove here, this assumption is not necessary.
Theorem 4.2
Let \(\alpha , \gamma , \theta \) be inner functions (\(\alpha ,\theta \) nonconstant). Let C be a conjugation in \(L^2\) such that \(M_z C=C M_{{{\bar{z}}}}\). Assume that \(C(\gamma K_\alpha )\subset K_\theta \). Then there is an inner function \(\beta \) such that \(C=C_\beta \), with \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta \) and \(\alpha \leqslant \theta \).
Proof
Recall the standard notation for the reproducing kernel function at 0 in \(K_\alpha \), namely, \(k_0^\alpha =1-\overline{\alpha (0)}{\alpha }\) and its conjugate \({{\tilde{k}}}_0^\alpha =C_\alpha k_0^\alpha =\bar{z}(\alpha -\alpha (0))\). By Theorem 2.2 we know that \(C=M_\psi J\) for some function \(\psi \in L^\infty \), \(|\psi |=1\). Hence
Thus there is \(h\in K_\theta \) such that \(h={{\bar{\gamma }}}{{\bar{\alpha }}} z\psi (1-\overline{\alpha (0)}\alpha )\). Since \((1-\overline{\alpha (0)}\alpha )^{-1}\) is a bounded analytic function, we have
Since \({{\bar{\gamma }}}{{\bar{\alpha }}} z\psi \in H^2\) and \(|\bar{\gamma }{{\bar{\alpha }}} z\psi |=1\) a.e. on \({\mathbb {T}}\), it has to be an inner function. Moreover \(\beta =z\psi \) has to be inner and divisible by \(\gamma \alpha \), i.e., \(\gamma \alpha \leqslant \beta \).
On the other hand, we have similarly
and \(\theta \gamma {{\bar{\beta }}}={\theta }{\gamma } {{\bar{z}}}{{\bar{\psi }}}\in H^2\). Hence \(\beta \) divides \( {\theta }{\gamma }\), i.e., \(\beta \leqslant \gamma \theta \). It is clear that \(C=C_\beta \). Finally, we have \(\alpha \leqslant \theta \) as a consequence of \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta \). \(\square \)
Note that if \(\alpha \leqslant \theta \) and \(C=C_\beta \) for some inner \(\beta \) with \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta \), then \(K_{\alpha }\subset K_{\tfrac{\beta }{\gamma }}\subset K_{\theta }\), \(C_{\beta }M_{\gamma }=C_{\tfrac{\beta }{\gamma }}\) and
Hence the implication in Theorem 4.2 is actually an equivalence.
The corollary bellow strengthens [2, Proposition 4.5].
Corollary 4.3
Let \(\alpha , \theta \) be nonconstant inner functions, and let C be a conjugation in \(L^2\) such that \(M_z C=C M_{{{\bar{z}}}}\). Assume that \(C(K_\alpha )\subset K_\theta \). Then \(\alpha \leqslant \theta \) and there is an inner function \(\beta \) such that \(C=C_\beta \), with \(\alpha \leqslant \beta \leqslant \theta \).
Let us turn to discussing the relations between \(M_z\)–commuting conjugations and model spaces. The following proposition describes some more properties of \(J^\star \).
Proposition 4.4
Let \(\alpha \) be an inner function. Then
-
(1)
\(J^\star (\alpha H^2)=\alpha ^{\#}H^2\);
-
(2)
\(J^\star (K_\alpha )=K_{\alpha ^{\#}}\);
-
(3)
\(J^{\star }C_\alpha =C_{\alpha ^{\#}}J^{\star }\).
Proof
The condition (1) is clear, (2) and (3) were proved in [4, Lemma 4.4]. \(\square \)
Hence the conjugation \(J^{\star }\) has a nice behaviour in connection with model spaces, namely \(J^{\star }(K_\alpha )=K_{\alpha ^{\#}}\). Theorem 4.6 below says that the conjugation \(J^{\star }\) is, in some sense the only \(M_z\)–commuting conjugation with this property.
We start with the following:
Proposition 4.5
Let \(\alpha , \gamma , \theta \) be inner functions (\(\alpha ,\theta \) nonconstant). Let C be an \(M_z\)–commuting conjugation in \(L^2\). Assume that \(C(\gamma K_\alpha )\subset K_\theta \). Then \(\alpha \leqslant \theta ^{\#}\) and there is an inner function \(\beta \) with \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta ^{\#}\) such that \(C=J^{\star } M_{\frac{\beta }{\gamma \alpha } {{\bar{\gamma }}}}\).
Proof
Observe that since C is an \(M_z\)–commuting conjugation, taking antilinear adjoints and applying [2, Proposition 2.1] we get \(M_{{{\bar{z}}}} C=C M_{{{\bar{z}}}}\). Since by Proposition 4.1 the antilinear operator \({M_{\gamma }} C_\alpha {M_{{\overline{\gamma }}}}\) is a conjugation, then \(J^{\star }C{M_{\gamma }} C_\alpha {M_{{\overline{\gamma }}}}\) is also a conjugation. Note also that \({M_{{\gamma }}} C_\alpha {M_{{\overline{\gamma }}}}M_z = M_{\bar{z}}{M_{{\gamma }}} C_\alpha {M_{{\overline{\gamma }}}}\). Hence
On the other hand,
By Theorem 4.2 there is an inner function \(\beta \) such that \( J^{\star }C {M_{{\gamma }}}C_\alpha {M_{{\overline{\gamma }}}}=C_\beta \), with \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta ^{\#}\) and \(\alpha \leqslant \theta ^{\#}\). Hence \(C=J^{\star } C_\beta {M_{{\gamma }}} C_\alpha {M_{{\overline{\gamma }}}}\). Therefore
\(\square \)
As in Theorem 4.2 the implication in Proposition 4.5 can be reversed. Indeed, if \(\alpha \leqslant \theta ^{\#}\) and \(C=J^{\star } M_{\frac{\beta }{\gamma \alpha } {\overline{\gamma }}}\) for some inner function \(\beta \) with \(\gamma \alpha \leqslant \beta \leqslant \gamma \theta ^{\#}\), then \(K_{\alpha }\subset K_{\tfrac{\beta }{\gamma }}\subset K_{\theta ^{\#}}\) and
Theorem 4.6
Let \(\alpha , \theta \) be nonconstant inner functions, and let C be an \(M_z\)–commuting conjugation in \(L^2\), i.e., \(M_z C=C M_{ z}\). Assume that \(C(K_\alpha )\subset K_\theta \). Then \(\alpha \leqslant \theta ^{\#}\) and \(C=\lambda J^{\star } \) with \(\lambda \in {\mathbb {T}}\).
Corollary 4.7
Let C be an \(M_z\)–commuting conjugation in \(L^2\). Assume that there is some nonconstant inner function \(\theta \) such that \(C(K_\theta )\subset K_{\theta ^{\#}}\). Then \(C=\lambda J^{\star }\) with \(\lambda \in {\mathbb {T}}\).
Proof of Theorem 4.6
By Proposition 4.5 there is an inner function \(\beta \) with \(\alpha \leqslant \beta \leqslant \theta ^{\#}\) such that \(C=J^{\star } M_{\frac{\beta }{ \alpha } }\). The function \(\frac{\beta }{ \alpha }\) is inner and by Theorem 2.4 it is symmetric. As observed before it follows that it is constant. Hence \(C=J^{\star } \) up to multiplication by a constant of modulus 1. \(\square \)
5 Conjugations preserving S-invariant subspaces of \(H^2\)
Beurling’s theorem says that all invariant subspaces for the unilateral shift S are of the form \(\theta H^2\) with \(\theta \) inner. We will now investigate conjugations in \(L^2\) which preserve subspaces of this form. Since \(C_{\theta }\) transforms \(\theta H^2\) onto \(\overline{z H^2}\), the operator
is an example of such a conjugation. Note that
Let \(\alpha \), \(\theta \) be two inner functions. Then the operator \(C_{\theta }J^{\star }C_{\alpha }\ :\ L^2\rightarrow L^2\) is an antilinear isometry which maps \(\alpha H^2\) onto \(\theta H^2\) and commutes with \(M_z\). This operator however does not have to be an involution.
Lemma 5.1
Let \(\alpha \), \(\theta \) be two inner functions. The operator \(C_{\theta }J^{\star }C_{\alpha }\) is an involution (and hence a conjugation in \(L^2\)) if and only if the function \(\theta \overline{\alpha ^{\#}}\) is symmetric (or equivalently \(\alpha \alpha ^{\#}=\theta \theta ^{\#}\)).
Proof
Note that by Proposition 4.1,
Therefore \(C_{\theta }J^{\star }C_{\alpha }\) is an involution if and only if
which means that
\(\square \)
The theorem bellow characterizes all \(M_z\)–commuting conjugations mapping one S–invariant subspace into another S–invariant subspace.
Theorem 5.2
Let \(\theta \) and \(\alpha \) be two inner functions and let C be a conjugation in \(L^2\) such that \(CM_z=M_zC\). Then \(C(\alpha H^2)\subset \theta H^2\) if and only if \(\theta \theta ^{\#}\leqslant \alpha \alpha ^{\#}\) and \(C=C_{\beta }J^{\star }C_{\alpha }\), where \(\beta \) is an inner function such that \(\theta \leqslant \beta \), \(\beta \beta ^{\#}=\alpha {\alpha ^{\#}}\). Moreover, in that case \(C(\alpha H^2)=\beta H^2\).
Let \(\alpha \) be a fixed inner function. By Lemma 5.1, for each inner function \(\beta \) with \(\beta \beta ^{\#}=\alpha \alpha ^{\#}\) there exists an \(M_z\)–commuting conjugation C which maps \(\alpha H^2\) onto \(\beta H^2\), namely \(C=C_{\beta }J^{\star }C_{\alpha }\). On the other hand, if \(\beta \) is an inner function and there exists an \(M_z\)–commuting conjugation C which maps \(\alpha H^2\) onto \(\beta H^2\), then by Theorem 5.2, \(\beta \beta ^{\#}\leqslant \alpha {\alpha ^{\#}}\) and \(C=C_{\gamma }J^{\star }C_{\alpha }\) for some inner function \(\gamma \) such that \(\beta \leqslant \gamma \), \(\gamma \gamma ^{\#}=\alpha \alpha ^{\#}\). In particular, \(C(\alpha H^2)=\gamma H^2=\beta H^2\) and so \(\gamma \) is a constant multiple of \(\beta \), \(\beta \beta ^{\#}=\alpha \alpha ^{\#}\).
It follows from the above that Lemma 5.3 characterizes all possible spaces of type \(\beta H^2\) such that for a given S–invariant subspace \(\alpha H^2\) there is an \(M_z\)–commuting conjugation mapping \(\alpha H^2\) onto \(\beta H^2\).
Lemma 5.3
Let \(\alpha \) be a nonconstant inner function. Then
For two inner functions \(\alpha \) and \(\beta \) denote by \(\alpha \wedge \beta \) the greatest common divisor of \(\alpha \) and \(\beta \). We will write \(\alpha \wedge \beta =1\) if the only common divisor of \(\alpha \) and \(\beta \) is a constant function.
Proof
Note that for \(\alpha =uv\) and \(\beta =\lambda uv^{\#}\) we have \(\alpha \alpha ^{\#}=\beta \beta ^{\#}\), hence one inclusion is proved. For the other inclusion let \(u=\alpha \wedge \beta \) and we can write \(\alpha =uv\) and \(\beta =u v_1\). From the condition \(\alpha \alpha ^{\#}=\beta \beta ^{\#}\) it follows that
Hence \(vv^{\#}=v_1v_1^{\#}\). Since \(v\wedge v_1=1\), we have that v divides \(v_1^{\#}\) and \(v_1\) divides \(v^{\#}\) and vice–versa. Thus we can take \(v_1=\lambda v^{\#}\) with \(\lambda \in {\mathbb {T}}\), and so \(\beta = \lambda uv^{\#}\). \(\square \)
Proof of Theorem 5.2
Assume firstly that \(CM_z=M_zC\) and \(C(\alpha H^2)\subset \theta H^2\). By Theorem 2.4, \(C=M_{\psi }J^{\star }\) for some unimodular symmetric function \(\psi \in L^{\infty }\). In particular,
and there exists \(u\in H^2\) such that \(\psi \alpha ^{\#}=\theta u\). Note that u must be inner and so \(\psi =\beta \overline{\alpha ^{\#}}\) with \(\beta =\theta u\), \(\theta \leqslant \beta \). Clearly \(\beta \overline{\alpha ^{\#}}\) is symmetric, i.e., \(\beta \beta ^{\#}=\alpha \alpha ^{\#}\). Hence \(\theta \theta ^{\#}\leqslant \alpha \alpha ^{\#}\).
Assume now that \(\theta \theta ^{\#}\leqslant \alpha \alpha ^{\#}\), and let \(\alpha = \alpha _1\cdot (\alpha \wedge \theta )\) and \(\theta =\theta _1\cdot (\alpha \wedge \theta )\). Since \((\alpha \wedge \theta )^{\#}=\alpha ^{\#}\wedge \theta ^{\#}\), we get \(\alpha ^{\#}= \alpha _1^{\#}\cdot (\alpha ^{\#}\wedge \theta ^{\#})\) and \(\theta ^{\#}=\theta _1^{\#}\cdot (\alpha ^{\#}\wedge \theta ^{\#})\). Note also that \(\theta _1\theta _1^{\#}\leqslant \alpha _1\alpha _1^{\#}\) and \(\theta _1\wedge \alpha _1=1\), so \(\theta _1\leqslant \alpha _1^{\#}\). Thus
where \(u=\frac{\alpha _1^{\#}}{\theta _1}\) is an inner function. Now we may take \(\beta =\theta u\). Since \(\theta \leqslant \beta \) and \(\beta \beta ^{\#}=\alpha {\alpha ^{\#}}\), by Lemma 5.1 and by Proposition 4.1, \(C=M_{\beta \overline{\alpha ^{\#}}}J^{\star }=C_{\beta }J^{\star }C_{\alpha }\) is a conjugation which maps \(\alpha H^2\) onto \(\beta H^2\subset \theta H^2\). \(\square \)
Corollary 5.4
Let \(\theta \) be an inner function and let C be an \(M_z\)–commuting conjugation in \(L^2\). Then
-
(1)
\(C(\theta H^2)\subset \theta H^2\) if and only if \(C=\lambda C_{\theta }J^{\star }C_{\theta }\) with \(\lambda \in {\mathbb {T}}\);
-
(2)
\(C(\theta H^2)\subset \theta ^{\#} H^2\) if and only if \(C=\lambda J^{\star }\) with \(\lambda \in {\mathbb {T}}\).
Proof
By Theorem 5.2, \(C(\theta H^2)\subset \theta H^2\) if and only if there exists an inner function \(\beta \) such that \(\theta \leqslant \beta \) and \(\beta \beta ^{\#}=\theta {\theta ^{\#}}\). This is only possible if \(\beta \) is constant multiple of \(\theta \) and (1) is proved. The proof of (2) is similar. \(\square \)
Note that by Theorem 5.2 (Lemma 5.1, actually) if \(\theta \overline{\alpha ^{\#}}\) is symmetric, then there exists an \(M_z\)–commuting conjugation from \(\alpha H^2\) into \(\theta H^2\). The following example shows that in that case there may be no such conjugation between the corresponding model spaces \(K_{\alpha }\) and \(K_{\theta }\).
Example 5.5
Fix \(a,b\in {\mathbb {D}}\) such that \(a\ne b\), \(a\ne {\overline{a}}\) and \(b\ne {\overline{b}}\), and put
Then
and so \(\alpha \alpha ^{\#}=\theta \theta ^{\#}\). Thus there exists an \(M_z\)–commuting conjugation from \(\alpha H^2\) onto \(\theta H^2\). In this case however neither \(\alpha \leqslant \theta ^{\#}\) nor \(\theta \leqslant \alpha ^{\#}\), so by Theorem 4.6 no \(M_z\)–commuting conjugation between \(K_{\alpha }\) and \(K_{\theta }\) exists. Here also neither \(\alpha \leqslant \theta \) nor \(\theta \leqslant \alpha \), and so by Theorem 4.2 no \(M_z\)–conjugation between \(K_{\alpha }\) and \(K_{\theta }\) exists.
Finally, consider \(M_z\)–conjugations preserving S–invariant subspaces.
Proposition 5.6
Let \(\theta \) and \(\alpha \) be two inner functions. There are no \(M_z\)–conjugations in \(L^2\) which map \(\alpha H^2\) into \(\theta H^2\).
Proof
If C was such a conjugation, then by Theorem 2.2, \(C=M_{\psi }J\) for some unimodular function \(\psi \in L^{\infty }\) and, in particular,
Clearly g must be an inner function and \(\psi =\alpha \theta g\). Then, for every \(h\in H^2\),
and so \(g{\overline{h}}\in H^2\). It follows that \(g=0\) and \(C(\alpha )=0\) which is a contradiction. \(\square \)
6 Conjugations and truncated Toeplitz operators
For \(\varphi \in L^2\) define the truncated Toeplitz operator\(A^\theta _\varphi \) by
were \(P_\theta :L^2\rightarrow K_\theta \) is the orthogonal projection (see [14]). The operator \(A^\theta _\varphi \) is closed and densely defined, and if it is bounded, it admits a unique bounded extension to \(K_\theta \). The set of all bounded truncated Toeplitz operators on \(K_\theta \) is denoted by \({\mathcal {T}}(\theta )\). Note that \(A^\theta _\varphi \in {\mathcal {T}}(\theta )\) for \(\varphi \in L^\infty \). It is known that every operator from \({\mathcal {T}}(\theta )\) is \(C_{\theta }\)–symmetric (see [14, Lemma 2.1]).
Observe that if \(k\geqslant 0\), then the conjugation \(C_{k,l}\) defined by (3.1) satisfies neither \(M_zC_{k,l}=C_{k,l}M_z\) nor \(SC_{k,l}=C_{k,l}S\). However, for \(0\leqslant n<k\) and \(\theta (z)=z^n\),
(since here \(C_{k,l|K_{\theta }}=J^{\star }_{|K_{\theta }}\) and \(\theta ^{\#}=\theta \)).
Theorem below characterizes conjugations intertwining truncated shifts \( A_z^{\theta }\) and \(A_z^{\theta ^{\#}}\).
Theorem 6.1
Let \(\theta \) be a nonconstant inner function and let C be a conjugation in \(L^2\) such that \(C(K_\theta )\subset K_{\theta ^{\#}}\). Then the following are equivalent:
-
(1)
\(A_{\varphi ^{\#}}^{\theta ^{\#}} C=C A_{\varphi }^{\theta }\) on \(K_{\theta }\) for all \(\varphi \in H^\infty \),
-
(2)
\(A_z^{\theta ^{\#}} C=C A_z^{\theta }\) on \(K_{\theta }\),
-
(3)
there is a function \(\psi \in H^{\infty }\) such that \(C_{|K_{\theta }}= J^{\star }A_{\psi }^{\theta }\) and \(A_{\psi }^{\theta }\) is an isometry,
-
(4)
there is a function \(\psi '\in H^{\infty }\) such that \(C_{|K_{\theta }}= A_{\psi '}^{\theta ^{\#}}J^{\star }_{|K_{\theta }}\) and \(A_{\psi '}^{\theta ^{\#}}\) is an isometry.
Proof
We will only prove that \((2)\Rightarrow (3)\). Since \(J^{\star }(K_{\theta ^{\#}})=K_\theta \) and \(J^{\star }A_{\varphi ^{\#}}^{\theta ^{\#}} = A_{\varphi }^{\theta }J^{\star }\) for all \(\varphi \in H^{\infty }\) (see [4, Lemma 4.5]), we have
on \(K_{\theta }\) and so \(J^{\star }C_{|K_{\theta }}=A_{\psi }^{\theta }\) for some \(\psi \in H^{\infty }\) ([5, Theorem 14.38]). Hence \(A_{\psi }^{\theta }\) is an isometry and
\(\square \)
It is much more restrictive if \(\theta ^{\#}=\theta \).
Proposition 6.2
Let \(\theta \) be an inner function such that \(\theta ^{\#}=\theta \) and let C be a conjugation in \(K_\theta \). Then the following are equivalent:
-
(1)
\(A_{\varphi ^{\#}}^{\theta }C=CA_{\varphi }^{\theta }\) for all \(\varphi \in H^\infty \),
-
(2)
\(A_z^{\theta } C= CA_z^{\theta }\),
-
(3)
\(C=\lambda J^{\star }_{|K_{\theta }}\) with \(\lambda \in {\mathbb {T}}\).
Proof
Implications \((1)\Rightarrow (2)\) and \((3)\Rightarrow (1)\) are clear. To prove \((2)\Rightarrow (3)\) apply Theorem 6.1 to the conjugation \({\tilde{C}}\) in \(L^2\) defined by
It follows that \({\tilde{C}}_{|K_{\theta }}=C=J^{\star }A_{\psi }^{\theta }\) for \(\psi \in H^{\infty }\) such that \(A_{\psi }^{\theta }\) is an isometry. Since \(C(K_{\theta })=K_{\theta }\) and \(J^{\star }(K_{\theta })=K_{\theta ^{\#}}=K_{\theta }\), we see that \(A_{\psi }^{\theta }=J^{\star }C\) maps \(K_{\theta }\) onto \(K_{\theta }\) and is in fact unitary. Thus we have
On the other hand, \(C^2=I_{K_{\theta }}\) so
Hence \(A_{{\overline{\psi }}}^{\theta }=A_{\psi ^{\#}}^{\theta }\) and \(A_{{\overline{\psi }}-\psi ^{\#}}^{\theta }=0\), which gives \({\overline{\psi }}-\psi ^{\#}\in \overline{\theta H^2}+\theta H^2\) (see [14]). In other words, \({\overline{\psi }}-\psi ^{\#}=\overline{\theta h_1}+\theta h_2\) for some functions \(h_1, h_2\in H^2\). Thus there exists a constant \(\lambda \) such that
We now have
Moreover \(\lambda \in {\mathbb {T}}\), since \(A_{\psi }^{\theta }\) is unitary. Hence
\(\square \)
Now we characterize conjugations intertwining the truncated shifts \( A_z^{\theta }\) and \(A_{{{\bar{z}}}}^{\theta }\).
Theorem 6.3
Let \(\theta \) be an inner function and let C be a conjugation in \(K_\theta \). Then the following are equivalent:
-
(1)
\(A_{\varphi }^{\theta } C=C A_{{\overline{\varphi }}}^{{\theta }}\) for all \(\varphi \in H^\infty \),
-
(2)
\(A_z^{\theta } C=C A_{{{\bar{z}}}}^{\theta }\) ,
-
(3)
there is a function \(\psi \in H^\infty \) such that \(C=A_\psi ^{\theta }C_{\theta }\) and \(A_\psi ^{\theta }\) is unitary,
-
(4)
there is a function \(\psi ^\prime \in H^\infty \) such that \(C=C_{\theta }A_{\overline{\psi ^\prime }}^{\theta }\) and \(A_{\psi ^\prime }^{\theta }\) is unitary.
Proof
Let us start with \((2)\Rightarrow (3)\). Since \(A_z^{\theta }\) is \(C_{\theta }\)–symmetric,
Hence, by [1, Proposition 1.21], \(CC_{\theta }=A_\psi ^{\theta }\) for some \(\psi \in H^\infty \). Clearly, \(A_\psi ^{\theta }\) is unitary and
To prove that \((4)\Rightarrow (1)\) note that, since \(A_{\overline{\psi ^\prime }}^{\theta }\) and \(A_{{\overline{\varphi }}}^{\theta }\) commute, we have
All other implications are straightforward. \(\square \)
Corollary 6.4
If C is a conjugation in \(K_\theta \) and every \(A\in {\mathcal {T}}(\theta )\) is C–symmetric, then \(C=A_\psi ^{\theta }C_{\theta }\) for some \(\psi \in H^\infty \) such that \(A_\psi ^{\theta }\) is unitary.
For a complete description of unitary operators from \({\mathcal {T}}(\theta )\) see [15, Proposition 6.5].
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The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2019 and the research of the second and the fourth authors was financed by the Ministry of Science and Higher Education of the Republic of Poland.
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Câmara, M.C., Kliś–Garlicka, K., Łanucha, B. et al. Conjugations in \(L^2\) and their invariants. Anal.Math.Phys. 10, 22 (2020). https://doi.org/10.1007/s13324-020-00364-5
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DOI: https://doi.org/10.1007/s13324-020-00364-5
Keywords
- Conjugation
- C–symmetric operator
- Hardy space
- Invariant subspaces for the unilateral shift
- Model space
- Truncated Toeplitz operator