Lower Semi-frames, Frames, and Metric Operators

Abstract

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

1 Introduction

In recent papers, one of us (RC) [21, 22] has analyzed sesquilinear forms defined by sequences in Hilbert spaces and operators associated with them by means of representation theorems. In particular, he derived results about lower semi-frames and duality.

It turns out that most results from [21, 22] can be extended to the continuous case and that is one of the aims of this present paper. The results are reported in Sect. 3, but we give here a brief summary. The continuous case involves a locally compact space $(X,\mu )$ with a Radon measure $\mu$. A function $\phi :X\rightarrow {\mathcal H}, x\mapsto \phi _x$ is said to be weakly measurable if for every $f\in {\mathcal H}$ the function $x\mapsto \langle {f} |{\phi _x} \rangle$ is measurable. A weakly measurable function $\phi$ is said to be $\mu$-total if $\langle {f} |{\phi _x} \rangle =0$ for a.e. $x\in X$ implies that $f=0$. A weakly measurable function $\phi$ is a continuous frame of ${\mathcal H}$ if there exist constants $0< \mathsf{m}\le \mathsf{M}<\infty$ (the frame bounds), such that:

$$\begin{aligned} \mathsf{m} \left\| f \right\| _{} ^2 \le \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) \le \mathsf{M} \left\| f \right\| _{} ^2 , \quad \forall \, f \in {\mathcal H}. \end{aligned}$$

(1.1)

If a weakly measurable function $\phi$ satisfies:

$$\begin{aligned} \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) \le { \mathsf M} \left\| f \right\| _{} ^2 , \quad \forall \, f \in {\mathcal H}, \, \end{aligned}$$

then we say that $\phi$ is a Bessel mapping of ${\mathcal H}$. On the other hand, $\phi$ is a lower semi-frame of ${\mathcal H}$ if there exists a constant $\mathsf{m}>0$, such that:

$$\begin{aligned} \mathsf{m} \left\| f \right\| _{} ^2 \le \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) , \quad \forall \, f \in {\mathcal H}. \end{aligned}$$

(1.2)

Given a weakly measurable function $\phi$, the operator $C_\phi :D(C_\phi )\subseteq {\mathcal H}\rightarrow L^2(X,d\mu )$ with domain

$$\begin{aligned} D(C_\phi ):=\left\{ f\in {\mathcal H}: \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) <\infty \right\} \end{aligned}$$

and $({C}_{\phi }f)(x) = \langle {f} |{\phi _{x}} \rangle$, $f \in D(C_\phi ),$ is called the analysis operator of $\phi$. We can define the sesquilinear form:

$$\begin{aligned} \Omega _\phi (f,g)=\int _X \langle {f} |{\phi _x} \rangle \langle {\phi _x} |{g} \rangle d\mu (x),\;\; f,g\in D(C_\phi ) \end{aligned}$$

and associate a positive self-adjoint operator $\mathsf{T}_\phi$ in the space ${\mathcal H}_\phi$, the closure of $D(C_\phi )$ in ${\mathcal H}$, by Kato’s representation theorem [26], which we call generalized frame operator. When $\phi$ is a lower semi-frame of ${\mathcal H}$, then the range of $\mathsf{T}_\phi$ is ${\mathcal H}_\phi$ and the function $\psi :X\rightarrow {\mathcal H}$, defined by $\psi _x=\mathsf{T}_\phi ^{-1}P_\phi \phi _x, x\in X$, where $P_\phi$ is the orthogonal projection onto ${\mathcal H}_\phi$, is a Bessel mapping and the reconstruction formula:

$$\begin{aligned} f=\int _X \langle {f} |{\phi _{x}} \rangle \psi _x \,\mathrm {d}\mu (x), \;\, \end{aligned}$$

holds for every $f\in D(C_\phi )$ in a weak sense.

In this paper, we do not confine ourselves to extend results of [21, 22], but actually we set two more goals. From one hand, given a lower semi-frame $\phi :X\rightarrow {\mathcal H}$ with $D(C_\phi )$ dense in ${\mathcal H}$, we consider general powers $\mathsf{T}_\phi ^{-k} \phi$ with $k\ge 0$. These functions are Bessel mappings, frames or lower semi-frames in the space ${\mathcal H}(\mathsf{T}_\phi ^m)$ (given by the domain of $\mathsf{T}_\phi ^m$ and the inner product $\langle {\mathsf{T}_\phi ^m \cdot } |{\mathsf{T}_\phi ^m \cdot } \rangle$) with $m\ge 0$ according to a simple relation between k and m (see Theorem 4.1).

When $\phi :X\rightarrow {\mathcal H}$ is a $\mu$-total weakly measurable function with $D(C_\phi )$ dense, then $\mathsf{T}_\phi$ is in particular a metric operator, i.e., a strictly positive self-adjoint operator. Metric operators are a topic familiar in the theory of the so-called ${\mathcal P}{\mathcal T}$-symmetric quantum mechanics [19, 27]. In our previous works [10, 11, 14], we have analyzed thoroughly the structure generated by such a metric operator, bounded or unbounded, namely a lattice of Hilbert spaces.

As particular case of Theorem 4.1, if $\phi :X\rightarrow {\mathcal H}$ is a lower semi-frame with $D(C_\phi )$ dense, then $\mathsf{T}_\phi ^{-1/2}\phi$ is a Parseval frame of ${\mathcal H}$. This inspired us to consider the following more general problem.

Question for which weakly measurable functions $\phi :X\rightarrow {\mathcal H}$, there exists a metric operator G on ${\mathcal H}$, such that $\phi _x\in D(G)$ for all $x\in X$ and $G\phi$ is a frame?

Partial answers to this problem are given in Theorem 6.1. In particular, necessary conditions are that $D(C_\phi )$ is dense, and that $\phi$ is $\mu$-total if $\phi$ is in addition a Bessel mapping. In the discrete case, if $\phi :\mathbb {N}\rightarrow {\mathcal H}$ is a Schauder basis, then the problem has a positive solution (more precisely, one can again take $G=T^{-1/2}_\phi$ and $T^{-1/2}_\phi \phi$ is actually an orthonormal basis). A particular case of this question has been treated, in the discrete case and with a different approach, in [25].

The paper is organized as follows. After reviewing the conventional definitions about frames and semi-frames in Sect. 2, we introduce in Sect. 3 the generalized frame operator $\mathsf{T}_\phi$, whose properties are more convenient that those of the standard frame operator $S_\phi$. In Sect. 4, we investigate the various (semi)-frames generated by a lower semi-frame. In Sect. 5, we review the lattice of Hilbert spaces generated by a metric operator. In Sect. 6, we face the question of transforming functions in frames. We conclude in Sect. 7 by several examples.

2 Preliminaries

Before proceeding, we list further definitions and conventions. The framework is a (separable) Hilbert space ${\mathcal H}$, with the inner product $\langle {\cdot } |{\cdot } \rangle$ linear in the first factor. $GL({\mathcal H})$ denotes the set of all invertible bounded operators on ${\mathcal H}$ with bounded inverse. Throughout the paper, we will consider weakly measurable functions $\phi : X \rightarrow {\mathcal H}$, where $(X,\mu )$ is a locally compact space with a Radon measure $\mu$.

Given a continuous frame $\phi$, the analysis operator is defined and bounded on ${\mathcal H}$, i.e., ${C}_{\phi }: \mathcal {H} \rightarrow L^{2}(X, \,\mathrm {d}\mu )$¹ and the corresponding synthesis operator ${C}_{\phi }^*: L^{2}(X, \,\mathrm {d}\mu ) \rightarrow {\mathcal H}$ is defined as (the integral being understood in the weak sense, as usual):

$$\begin{aligned} {C}_{\phi }^*\xi = \int _X \xi (x) \,\phi _{x} \; \,\mathrm {d}\mu (x),\quad \text{ for } \;\;\xi \in L^{2}(X, \,\mathrm {d}\mu ). \end{aligned}$$

(2.1)

Moreover, we set $S_\phi :={C}_{\phi }^* {C}_{\phi }$, which is self-adjoint. Then, it follows that:

$$\begin{aligned} \langle {S_\phi f} |{g} \rangle = \langle {{C}^*_\phi {C}_\phi f} |{g} \rangle = \langle { {C}_\phi f} |{{C}_\phi g} \rangle = \int _{X} \langle {f} |{\phi _{x}} \rangle \langle {\phi _x} |{g} \rangle \, \,\mathrm {d}\mu (x) . \end{aligned}$$

Thus, for continuous frames, $S_\phi$ and $S_\phi ^{-1}$ are both bounded, that is, $S_\phi \in GL({\mathcal H})$.

Following [6, 7], we will say that a function $\phi$ is a semi-frame if it satisfies only one of the frame inequalities in (1.1). We already introduced the lower semi-frames (if $\phi$ satisfies (1.2), then it could still be a frame, and hence, we say that the lower semi-frame $\phi$ is proper if it is not a frame). Note that the lower frame inequality automatically implies that $\phi$ is $\mu$-total. On the other hand, a weakly measurable function $\phi$ is an upper semi-frame if is $\mu$-total, that is, $N(C_{\phi } )= \{0\}$, and there exists $\mathsf{M}<\infty$, such that:

$$\begin{aligned} 0 < \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) \le { \mathsf M} \left\| f \right\| _{} ^2 , \quad \forall \, f \in {\mathcal H}, \, f\ne 0. \end{aligned}$$

(2.2)

Thus, an upper semi-frame is a total Bessel mapping [24]. Notice that this definition does not forbid $\phi$ to be a frame. Thus, we say that $\phi$ is a proper upper semi-frame if it is not a frame.

If $\phi$ is a proper upper semi-frame, $S_\phi$ is bounded and $S_\phi ^{-1}$ is unbounded, as follows immediately from (2.2). In the lower case, however, the definition of $S_\phi$ must be changed, since the domain $D(C_\phi )$ need not be dense, so that ${C}_{\phi }^*$ may not exist. Instead, following [6, Sec. 2], one defines the synthesis operator as:

$$\begin{aligned} {D}_{\phi }F = \int _X F(x) \,\phi _{x} \; \,\mathrm {d}\mu (x), \quad F\in L^{2}(X, \,\mathrm {d}\mu ) , \end{aligned}$$

(2.3)

on the domain of all elements F for which the integral in (2.3) converges weakly in ${\mathcal H}$, and then, $S_\phi :={D}_{\phi } {C}_{\phi }$. With this definition, it is shown in [6, Sec. 2] that if $\phi$ is a proper lower semi-frame, then $S_\phi$ is unbounded and $S_\phi ^{-1}$ is bounded. All these objects are studied in detail in our previous papers [6, 7]. In particular, it is shown there that a natural notion of duality exists, namely, two measurable functions $\phi , \psi$ are dual to each other (the relation is symmetric) if one has:

$$\begin{aligned} \langle {f} |{g} \rangle = \int _X \langle {f} |{\phi _{x}} \rangle \langle {\psi _{x}} |{g} \rangle \, \; \,\mathrm {d}\mu (x), \quad \forall \, f,g \in {\mathcal H}. \end{aligned}$$

(2.4)

This duality property extends to lower semi-frames and Bessel mappings, as shown in Proposition 3.2 below.

Consider the following sesquilinear form on the domain ${\mathcal D}_1 \times {\mathcal D}_2$ :

$$\begin{aligned} \Omega _{\psi , \phi }(f,g) = \int _X \langle {f} |{\psi _x} \rangle \langle {\phi _x} |{g} \rangle \,\mathrm {d}\mu (x), \quad f\in {\mathcal D}_1, g\in {\mathcal D}_2. \end{aligned}$$

(2.5)

If ${\mathcal D}_1 ={\mathcal D}_2= {\mathcal H}$ and the form $\Omega _{\psi , \phi }$ is bounded on ${\mathcal H}\times {\mathcal H}$, that is, $| \Omega _{\psi , \phi }(f,g) | \le c \left\| f \right\| _{} \left\| g \right\| _{}$, for some $c>0$, then the couple of weakly measurable functions $(\psi , \phi )$ is called a reproducing pair if the corresponding bounded operator $S_{\psi , \phi }$ given weakly by:

$$\begin{aligned} \langle {S_{\psi , \phi }f} |{g} \rangle = \int _X \langle {f} |{\psi _x} \rangle \langle {\phi _x} |{g} \rangle \,\mathrm {d}\mu (x) , \quad \forall \,f, g\in {\mathcal H}, \end{aligned}$$

belongs to $GL({\mathcal H})$. If $\psi = \phi$, we recover the notion of continuous frame.

Under certain conditions, boundedness of $\Omega _{\psi , \phi }$ is automatic, as shown in [21, Prop. 7].

Proposition 2.1

If ${\mathcal D}_1 ={\mathcal D}_2= {\mathcal H}$, X is locally compact and $\sigma$-compact, that is, $X = \bigcup _{n} K_{n}, K_{n} \subset K_{n+1}$, with $K_{j}$ compact for every j, and $\sup _{x\in X}( \left\| \phi _x \right\| _{{\mathcal H}}$$\left\| \psi _x \right\| _{{\mathcal H}} ) <\infty$, then the form $\Omega _{\psi , \phi }$ is bounded on ${\mathcal H}\times {\mathcal H}$.

Proof

Define:

$$\begin{aligned} \Omega ^n_{\psi , \phi }(f,g) := \int _{K_n} \langle {f} |{\psi _x} \rangle \langle {\phi _x} |{g} \rangle \,\mathrm {d}\mu (x), \quad f,g \in {\mathcal H}. \end{aligned}$$

By assumption, there exists $c>0$, such that:

$$\begin{aligned} |\Omega ^n_{\psi , \phi }(f,g)| \le c \left\| f \right\| _{} \left\| g \right\| _{} \left| \int _{K_n} \,\mathrm {d}\mu (x) \right| < \infty . \end{aligned}$$

Hence, there exists a bounded operator $T_n$, such that $\Omega ^n_{\psi , \phi }(f,g) = \langle {T_n f} |{g} \rangle$. Applying the Banach–Steinhaus theorem to the functional $g\mapsto \Omega _{\psi , \phi }(f,g)$, one gets that the operator $T_{\psi , \phi }$ associated with $\Omega _{\psi , \phi }$ is defined on the whole of ${\mathcal H}$. Doing the same with $T_n^*$, as in [21, Prop. 7], we conclude that the form $\Omega _{\psi , \phi }$ is bounded on ${\mathcal H}\times {\mathcal H}$. $\square$

The converse of Proposition 2.1 does not hold, as shown in the following example (which is based on [17, Example 2.5]), in the sense that boundedness of the form $\Omega$ does not imply the supremum condition.

Example 2.2

Let $h\in {\mathcal H}\backslash \{0\}$ and let $a:\mathbb {R}\rightarrow {\mathbb C}$ be such that $a\in L^2(\mathbb {R})\backslash L^\infty$$(\mathbb {R})$. Define $\phi _x=a(x)h$ for all $x\in \mathbb {R}$. Then, $\phi :\mathbb {R}\rightarrow {\mathcal H}$ is a weakly measurable function and a Bessel mapping, because:

$$\begin{aligned} \int _\mathbb {R}| \langle {f} |{\phi _x} \rangle |^2 dx \le \left\| a \right\| _{L^2(\mathbb {R})} ^2 \Vert h\Vert ^2\Vert f\Vert ^2, \quad f\in {\mathcal H}. \end{aligned}$$

In conclusion, the sesquilinear form $\Omega _{\psi , \phi }$, where $\psi =\phi$, is defined and bounded on ${\mathcal H}\times {\mathcal H}$, but $\sup _{x\in X}( \left\| \phi _x \right\| _{{\mathcal H}} \left\| \psi _x \right\| _{{\mathcal H}} ) =\infty$.

3 The Generalized Frame Operator $\mathsf{T}_\phi$

In the previous section, we defined the frame operator $S_\phi$ for a lower semi-frame. However, this operator lacks good properties, in general (for instance $S_\phi$ need not be self-adjoint like in the case of an upper semi-frame, even if $S_\phi$ is non-negative). In this section, we are going to construct a new operator associated with $\phi$ which plays the rôle of $S_\phi$ for lower semi-frames. We show its main properties, in particular, concerning the definition of a Bessel dual mapping in a natural way.

We note that if $\phi$ is a proper lower semi-frame, the r.h.s. of (1.2) actually diverges for some f. As already said, the domain $D(C_\phi )$ need not be dense in ${\mathcal H}$. It is useful to work with the Hilbert space ${\mathcal H}_\phi$ made of the closure of $D(C_\phi )$ endowed with the topology of ${\mathcal H}$.

The analysis operator $C_\phi$ is closed [6, Lemma 2.1]. Therefore, the sesquilinear form:

$$\begin{aligned} \Omega _\phi (f,g)={ \langle { {C}_\phi f} |{{C}_\phi g} \rangle }=\int _X \langle {f} |{\phi _x} \rangle \langle {\phi _x} |{g} \rangle d\mu (x),\quad f,g\in D(C_\phi ) \end{aligned}$$

is non-negative and closed. By Kato’s second representation theorem [26, Theorem 2.23], there exists an operator $\mathsf{T}_\phi :D(\mathsf{T}_\phi )\subset {\mathcal H}_\phi \rightarrow {\mathcal H}_\phi$, with $D(\mathsf{T}_\phi )\subset D(C_\phi )$, such that²

• $\Omega _\phi (f,g)= \langle {\mathsf{T}_\phi f} |{g} \rangle$ for all $f\in D(\mathsf{T}_\phi )$ and $g\in D(C_\phi )$;

• if $T:D(T)\subset {\mathcal H}_\phi \rightarrow {\mathcal H}_\phi$ is such that $\Omega _\phi (f,g)= \langle {T f} |{g} \rangle$ for all $f\in D(T)$ and $g\in D(C_\phi )$, then $T\subset \mathsf{T}_\phi$;

• $\mathsf{T}_\phi$ is non-negative and self-adjoint in ${\mathcal H}_\phi$;

• $\Omega _\phi (f,g)= \langle {\mathsf{T}_\phi ^{1/ 2} f} |{\mathsf{T}_\phi ^{1/2} g} \rangle$, for all $f,g\in D(C_\phi )$.

We call $\mathsf{T}_\phi$ the generalized frame operator of $\phi$. The motivation behind this name is that when $\phi$ is a continuous frame, then $\mathsf{T}_\phi =S_\phi$. The generalized frame operator has been studied in [21, 22] in the discrete case and a preliminary extension to the continuous setting has been given in [18].

If $D(C_\phi )$ is dense, then, of course, ${\mathcal H}_\phi ={\mathcal H}$ (i.e., $\mathsf{T}_\phi$ is a non-negative self-adjoint operator on ${\mathcal H}$) and $S_\phi \subset \mathsf{T}_\phi$. In [22], it was proved that in the discrete setting ($X=\mathbb {N}$) we may have a strict inclusion $S_\phi \subsetneq \mathsf{T}_\phi$ (recall that in our paper $S_\phi$ is weakly defined; therefore, in the discrete setting, it corresponds to the operator $W_\phi$ in [22]).

Since $\Omega _\phi (f,g)= \langle {C_\phi f} |{C_\phi g} \rangle$, we have $\mathsf{T}_\phi :=|C_\phi |^2=C_\phi ^\times C_\phi$, where $C_\phi ^\times$ and $|C_\phi |$ are the adjoint and the modulus of $C_\phi$ when we think of it as an operator $C_\phi :{\mathcal H}_\phi \rightarrow L^2(X,\mu )$.

The following characterization can be proved as in [21].

Proposition 3.1

Let $\phi$ be a weakly measurable function and $\mathsf{m}>0$. The following statements are equivalent.

• (i) $\phi$ is a lower semi-frame of ${\mathcal H}$ with lower bound m;

• (ii) $\Omega _\phi$ is bounded from below by m, that is:

  $$\begin{aligned} \Omega _\phi (f,f)\ge \mathsf{m}\Vert f\Vert ^2,\quad \forall f\in D(C_\phi ); \end{aligned}$$

• (iii) $C_\phi$ is bounded from below by $\sqrt{\mathsf{m}}$, that is:

  $$\begin{aligned} \Vert C_\phi f\Vert \ge \sqrt{\mathsf{m}}\Vert f\Vert ,\quad \forall f\in D(C_\phi ); \end{aligned}$$

• (iv) $\mathsf{T}_\phi$ is bounded from below by m, that is:

  $$\begin{aligned} \Vert \mathsf{T}_\phi f\Vert \ge \mathsf{m}\Vert f\Vert ,\quad \forall f\in D(\mathsf{T}_\phi ); \end{aligned}$$

• (v) $\mathsf{T}_\phi$ is invertible and $\mathsf{T}_\phi ^{-1}\in {\mathcal B}({\mathcal H}_\phi )$ with $\Vert \mathsf{T}_\phi ^{-1}\Vert \le \mathsf{m}$.

Proof

(i) $\iff$ (ii) and (i) $\iff$ (iii) By definition:

(ii) $\implies$ (iv) Let $f\in D(\mathsf{T}_\phi )$. Then, $\Vert \mathsf{T}_\phi f\Vert \Vert f\Vert \ge \langle {\mathsf{T}_\phi f} |{f} \rangle \ge \mathsf{m}\Vert f\Vert ^2$. Thus $\Vert \mathsf{T}_\phi f\Vert \ge \mathsf{m}\Vert f\Vert$.

(iv) $\implies$ (ii) For $f\in D(\mathsf{T}_\phi )$, we have $\Omega _\phi (f,f)= \langle {\mathsf{T}_\phi ^{1/2} f} |{\mathsf{T}_\phi ^{1/2} f} \rangle \ge \mathsf{m}\Vert f\Vert ^2$ by hypothesis. The inequality now extends to every $f\in D(C_\phi )$ noting that $D(\mathsf{T}_\phi )$ is a core of $\Omega _\phi$ (see [26, Theorem 2.1]).

(iv) $\iff$ (v) This is clear, since $\mathsf{T}_\phi$ is self-adjoint. $\square$

Now, assume that $\phi$ is a lower semi-frame of ${\mathcal H}$. By [6, Proposition 2.6], there exists a Bessel mapping $\psi :X\rightarrow {\mathcal H}$ dual to $\phi$, i.e., for some $\mathsf{M}>0$:

$$\begin{aligned} \int _{X} | \langle {f} |{\psi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) \le \mathsf{M} \left\| f \right\| _{} ^2 , \quad \forall \, f \in {\mathcal H}, \end{aligned}$$

and

$$\begin{aligned} f=\int _X \langle {f} |{\phi _{x}} \rangle \psi _x \,\mathrm {d}\mu (x), \quad \text {for all } f\in D(C_\phi ) \text { in weak sense. } \end{aligned}$$

(3.1)

A proof may be found in [6, Lemma 2.5 and Proposition 2.6]. Note that the proof given there is incomplete, in the sense, there is no guarantee that $\psi$ is total, see Item (4) in Proposition 3.2 below. Here, we give a different proof involving the operator $\mathsf{T}_\phi$ using the same argument as in [22, Theorem 4.1].

Proposition 3.2

Let $\phi$ be a lower semi-frame of ${\mathcal H}$. Let $\psi$ be the function defined by $\psi =\mathsf{T}_\phi ^{-1}P_\phi \phi$, i.e., $\psi _x=\mathsf{T}_\phi ^{-1}P_\phi \phi _x$ for $x\in X$, where $P_\phi :{\mathcal H}\rightarrow {\mathcal H}_\phi$ is the orthogonal projection of ${\mathcal H}$ onto ${\mathcal H}_\phi$. The following statements hold.

• (1) $\psi$ is a weakly measurable function;

• (2) $\psi$ a Bessel mapping of ${\mathcal H}$;

• (3) $\psi$ is dual to $\phi$, i.e., satisfies (3.1).

• (4) If $D(C_\phi )$ is dense, then $\psi$ is an upper semi-frame of ${\mathcal H}$.

Proof

We recall that $\mathsf{T}_\phi$ is invertible with $\mathsf{T}_\phi ^{-1}\in {\mathcal B}({\mathcal H}_\phi )$.

• (1) This is immediate.

• (2) Let $f\in {\mathcal H}$. Then:

  $$\begin{aligned} \int _{X} | \langle {f} |{\mathsf{T}_\phi ^{-1}P_\phi \phi _x} \rangle |^2 \,\mathrm {d}\mu (x)= & {} \int _{X} | \langle {P_\phi f} |{\mathsf{T}_\phi ^{-1}P_\phi \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \int _{X} | \langle {\mathsf{T}_\phi ^{-1} P_\phi f} |{ \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \Vert \mathsf{T}_\phi ^{1/2} \mathsf{T}_\phi ^{-1} P_\phi f \Vert ^2\nonumber \\= & {} \Vert \mathsf{T}_\phi ^{-1/2} P_\phi f \Vert ^2 {\le } \Vert \mathsf{T}_\phi ^{-1/2}\Vert ^2 \Vert f \Vert ^2. \end{aligned}$$

  (3.2)

• (3) Let $f\in D(C_\phi )$ and $h\in {\mathcal H}$. Then:

  $$\begin{aligned} \langle {f} |{h} \rangle= & {} \langle {f} |{P_\phi h} \rangle = \langle {f} |{\mathsf{T}_\phi \mathsf{T}_\phi ^{-1}P_\phi h} \rangle =\int _X \langle {f} |{\phi _x} \rangle \langle {\phi _x} |{\mathsf{T}_\phi ^{-1}P_\phi h} \rangle \,\mathrm {d}\mu (x)\\= & {} \int _X \langle {f} |{\phi _x} \rangle \langle {P_\phi \phi _x} |{\mathsf{T}_\phi ^{-1}P_\phi h} \rangle \,\mathrm {d}\mu (x)= \int _X \langle {f} |{\phi _x} \rangle \langle {\mathsf{T}_\phi ^{-1}P_\phi \phi _x} |{h} \rangle \,\mathrm {d}\mu (x). \end{aligned}$$

• (4) This is immediate from (3.1). $\square$

Remark 3.3

Part (3) of the above proposition comes from general properties of semi-bounded operators. If ${\mathcal H}, {\mathcal K}$ are Hilbert spaces, and $C:D(C)\subset {\mathcal H}\rightarrow {\mathcal K}$ is a closed operator, such that D(C) is dense and $\Vert Cf\Vert \ge \gamma \Vert f\Vert$, for every $f\in D(C)$, then by Kato’s theorem, there exists $T\ge 0$, with the properties described at the beginning of Sect. 3, such that $\langle {Cf} |{Cg} \rangle = \langle {T^{1/2}f} |{T^{1/2}g} \rangle$ for every $f,g \in D(C)$; as before T is invertible with bounded inverse. On the other hand:

$$\begin{aligned} | \langle {f} |{g} \rangle | \le \frac{1}{\gamma ^2} \Vert f\Vert _C \Vert g\Vert _C, \quad \forall f,g \in D(C),\;\; \text{ where } \;\Vert f\Vert _C: = \left\| Cf \right\| _{} . \end{aligned}$$

Hence, there exists an operator X, bounded in ${\mathcal H}(C):=D(C)[\Vert \cdot \Vert _C]$, such that:

$$\begin{aligned} \langle {f} |{g} \rangle = \langle {Cf} |{CXg} \rangle , \quad \forall f,g \in D(C). \end{aligned}$$

Then:

$$\begin{aligned} \langle {Cf} |{CXg} \rangle = \langle {T^{1/2}f} |{T^{1/2}Xg} \rangle = \langle {f} |{g} \rangle , \quad \forall f,g \in D(C). \end{aligned}$$

This implies that $T^{1/2}Xg \in D(T^{1/2})=D(C)$, for every $g\in D(C)$ and $TXg=g$, $\forall g\in D(C)$. Thus, $X\subset T^{-1}$. In conclusion, $\langle {f} |{g} \rangle = \langle {Cf} |{CT^{-1}g} \rangle$, for all $f,g \in D(C)$. Therefore, if $C=C_\phi$ is densely defined, the dual is found. If $D(C_\phi )$ is not dense, we can proceed with the projection $P_\phi$ as before.

Finally, as mentioned in [22] for the discrete case, calculations similar to (3.2) show the next result.

Proposition 3.4

Let $\phi$ be a lower semi-frame of ${\mathcal H}$. Then, the function $\psi = \mathsf{T}_\phi ^{-1/2}P_\phi \phi$ is a Parseval frame for ${\mathcal H}_\phi$.

Thus, following the standard terminology in frame theory, we can call $\mathsf{T}_\phi ^{-1}P_\phi \phi$ and $\mathsf{T}_\phi ^{-1/2}P_\phi \phi$ the canonical dual Bessel mapping and the canonical tight frame of $\phi$.

Now, we exploit Proposition 5 of [21] and the discussion after it.

Proposition 3.5

Let $\phi$ be a weakly measurable function of ${\mathcal H}$. Then, $\phi$ is a lower semi-frame of ${\mathcal H}$ if and only if there exists an inner product $\langle {\cdot } |{\cdot } \rangle _+$ inducing a norm $\left\| \cdot \right\| _{+}$ on $D(C_\phi )$ for which $D(C_\phi )[ \left\| \cdot \right\| _{+} ]$ is complete, continuously embedded into ${\mathcal H}$, and for some $\alpha ,\mathsf{m}, \mathsf{M} > 0$, one has:

$$\begin{aligned}&\alpha \left\| f \right\| _{} \le \left\| f \right\| _{+} \; \text {and}\\&\mathsf{m} \left\| f \right\| _{+} ^2 \le \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x) \le \mathsf{M} \left\| f \right\| _{+} ^2 , \quad \forall \, \in D(C_\phi ). \end{aligned}$$

Proof

It is sufficient to take $\left\| f \right\| _{+} ^2 = \left\| f \right\| _{C_\phi } ^2= \int _{X} | \langle {f} |{\phi _{x}} \rangle | ^2 \, \,\mathrm {d}\mu (x),$ for $f\in D(C_\phi )$. $\square$

Let $\phi$ be a lower semi-frame in the Hilbert space ${\mathcal H}$, with domain $D(C_\phi )$, assumed to be dense. For every $x\in X$, the map $f\mapsto \langle {f} |{\phi _x} \rangle$ is a bounded linear functional on the Hilbert space ${\mathcal H}(C_\phi ):= D(C_\phi )[ \left\| \cdot \right\| _{+} ]$. By the Riesz Lemma, there exists an element $\chi _x^{\phi } \in D(C_\phi )$, such that:

$$\begin{aligned} \langle {f} |{\phi _x} \rangle = \langle {f} |{\chi _x^{\phi }} \rangle _+ \,\,\, \forall \, f \in D(C_\phi ). \end{aligned}$$

(3.3)

By Proposition 3.5, $\chi ^{\phi }$ is a frame.

We can explicitly determine the element $\chi ^{\phi }$ when $\left\| \cdot \right\| _{+}$ is the norm $\left\| f \right\| _{1/2} ^2 =\Vert {\mathsf{T}_\phi ^{1/2}f}^2\Vert$. Notice that, by Proposition 3.1(iv), this norm is equivalent to the graph norm of $\mathsf{T}_\phi ^{1/2}$.

Then, we have $\langle {f} |{\phi _x} \rangle = \langle {f} |{\chi _x^{\phi }} \rangle _{1/2}= \langle { f} |{\mathsf{T}_\phi \chi _x^{\phi }} \rangle$ for all $f\in D(C_\phi )$. Thus $\chi _x^{\phi }=\mathsf{T}_\phi ^{-1} \phi _x$ for all $x\in X$, i.e., $\chi ^\phi$ is the canonical dual Bessel mapping of $\phi$.

Following the notation of [8], denote by ${\mathcal H}( \mathsf{T}_\phi ^{1/2})$ the Hilbert space $D(\mathsf{T}_\phi ^{1/2})$ with the norm $\left\| \cdot \right\| _{1/2}$. In the same way, denote by ${\mathcal H}( C_\phi )$ the Hilbert space $D(C_\phi )$ with inner product $\langle {C_\phi \cdot } |{C_\phi \cdot } \rangle$. Hence, we have proved the following result.

Proposition 3.6

Let $\phi$ be a lower semi-frame of ${\mathcal H}$ with $D(C_\phi )$ dense. Then, the canonical dual Bessel mapping of $\phi$ is a tight frame for the Hilbert space ${\mathcal H}(C_\phi )= {\mathcal H}( \mathsf{T}_\phi ^{1/2})$.

We can proceed in the converse direction, i.e., starting with a frame $\chi \in D(C_\phi )$, does there exist a lower semi-frame $\eta$ of ${\mathcal H}$, such that $\chi$ is the frame $\chi ^{\eta }$ constructed from $\eta$ in the way described above? The answer is formulated in the following.

Proposition 3.7

[21, Prop. 6] Let $\chi$ be a frame of ${\mathcal H}(C_\phi )={\mathcal H}(\mathsf{T}_\phi ^{1/2} )$. Then:

• (i) There exists a lower semi-frame $\eta$ of ${\mathcal H}$, such that $\chi = \chi ^{\eta }$ if, and only if, $\chi \in D(\mathsf{T}_\phi )$.

• (ii) If $\chi =\chi ^{\eta }$ for some lower semi-frame $\eta$ of ${\mathcal H}$, then $\eta = \mathsf{T}_\phi \chi$.

4 The Functions Generated by a Lower Semi-frame

Throughout this section, we continue to consider a lower semi-frame $\phi$ in ${\mathcal H}$, with $D(C_\phi )$ dense.

Since $\mathsf{T}_\phi ^{-1}$ is defined on ${\mathcal H}$, we can actually apply different powers of $\mathsf{T}_\phi ^{-1}$ on $\phi$ and get the functions $\mathsf{T}_\phi ^{-k} \phi$, $k\in [0,\infty )$. Hence, we can ask for the properties of $\mathsf{T}_\phi ^{-k} \phi$. Of course, the answer depends on the Hilbert space where $\mathsf{T}_\phi ^{-k} \phi$ is considered. For instance, for $k=0$, we have a lower semi-frame of ${\mathcal H}$ and, as seen in the previous section, for $k=\frac{1}{2}$, we have a frame for ${\mathcal H}$, while for $k=1$, we have a frame for ${\mathcal H}( \mathsf{T}_\phi ^{1/2})$. When we have powers of an unbounded, closed, densely defined operator, then we can consider scales and lattices of Hilbert spaces, which we will consider in more detail in Sect. 5. For a while, let us simply denote by ${\mathcal H}(\mathsf{T}_\phi ^m)$, $m\ge 0$, the domain of $\mathsf{T}_\phi ^m$ considered as a Hilbert space with norm $\Vert f\Vert _m= \Vert \mathsf{T}_\phi ^m f\Vert$, $f\in D(\mathsf{T}_\phi ^m)$. Then, if $m>n\ge 0$, we have ${\mathcal H}(\mathsf{T}_\phi ^m) \subset {\mathcal H}(\mathsf{T}_\phi ^n) \subset {\mathcal H}$. The function $\mathsf{T}_\phi ^{-k} \phi$ is not always a well-defined function of ${\mathcal H}(\mathsf{T}_\phi ^{m})$; a sufficient condition for $\mathsf{T}_\phi ^{-k} \phi \in {\mathcal H}(\mathsf{T}_\phi ^{m}$) is that $k\ge {m}$.

Having at our disposal the notion of scale of Hilbert spaces, we now come back to a lower semi-frame $\phi$ with $D(C_\phi )$ dense and to the functions $\mathsf{T}_\phi ^{-k}\phi$ with $k\ge 0$. For simplicity of notation, we write in a compact way the inner product of ${\mathcal H}(\mathsf{T}_\phi ^m)$ in the following way $\langle {f} |{g} \rangle _m:= \langle {f} |{g} \rangle _{\mathsf{T}_\phi ^m}= \langle {\mathsf{T}_\phi ^m f} |{\mathsf{T}_\phi ^m g} \rangle$.

Theorem 4.1

Let $\phi$ be a lower semi-frame of ${\mathcal H}$ with $D(C_\phi )$ dense and $m,k\in [0,\infty )$, $k\ge m$. Then, the following statements hold.

• (i) $\mathsf{T}_\phi ^{-k}\phi$ is a Bessel mapping of ${\mathcal H}(\mathsf{T}_\phi ^m)$ if and only if $k\ge m+\frac{1}{2}$;

• (ii) $\mathsf{T}_\phi ^{-k}\phi$ is a lower semi-frame of ${\mathcal H}(\mathsf{T}_\phi ^m)$ if and only if $m\le k \le m+\frac{1}{2}$;

• (iii) $\mathsf{T}_\phi ^{-k}\phi$ is a frame of ${\mathcal H}(\mathsf{T}_\phi ^m)$ if and only if $\mathsf{T}_\phi ^{-k}\phi$ is a Parseval frame of ${\mathcal H}(\mathsf{T}_\phi ^m)$, if and only if $k= m+\frac{1}{2}$.

Proof

We have for $f\in {\mathcal H}(\mathsf{T}_\phi ^m)$:

$$\begin{aligned} \int _{X} | \langle {f} |{\mathsf{T}_\phi ^{-k} \phi _x} \rangle |_{m}^2 \,\mathrm {d}\mu (x)= & {} \int _{X} | \langle {\mathsf{T}_\phi ^{m} f} |{\mathsf{T}_\phi ^{m}\mathsf{T}_\phi ^{-k} \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \int _{X} | \langle {\mathsf{T}_\phi ^{2m-k} f} |{ \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \Vert \mathsf{T}_\phi ^{1/2} \mathsf{T}_\phi ^{2m-k} f \Vert ^2= \Vert \mathsf{T}_\phi ^{2m-k+1/2} f \Vert ^2. \end{aligned}$$

(4.1)

Hence, taking into account that $0\in \rho (\mathsf{T}_\phi )$ and that $\Vert f\Vert ^2_{\mathsf{T}_\phi ^m}=\Vert \mathsf{T}_\phi ^m f\Vert ^2$, (i) and (ii) follow by comparing $2m-k+\frac{1}{2}$ and m. We give the details for (i) as example.

Assume that $\mathsf{T}_\phi ^{-k}\phi$ is a Bessel mapping of ${\mathcal H}(\mathsf{T}_\phi ^m)$, then there exists $B>0$, such that for all $f\in {\mathcal D}(\mathsf{T}_\phi ^m)$, we have $\Vert \mathsf{T}_\phi ^{2m-k+1/2} f \Vert ^2\le B \Vert \mathsf{T}_\phi ^m f\Vert ^2$. Rewriting the inequality as $\Vert \mathsf{T}_\phi ^{m-k+\frac{1}{2}} \mathsf{T}_\phi ^{m} f \Vert ^2\le B \Vert \mathsf{T}_\phi ^m f\Vert ^2$, we find that $\mathsf{T}_\phi ^{m-k+\frac{1}{2}}$ is bounded, i.e., $m-k+\frac{1}{2}\le 0$. The other implication trivially holds by the same estimate.

Finally, combining the two cases above, we obtain that $\mathsf{T}_\phi ^{-k}\phi$ is a frame of ${\mathcal H}(\mathsf{T}_\phi ^m)$ if and only if $k= m+\frac{1}{2}$. Moreover, in this case, $\mathsf{T}_\phi ^{-k}\phi$ is actually a Parseval frame, as one can see in (4.1). $\square$

In particular, we recover some cases we discussed in the previous section, namely $(k,m)=(1,\frac{1}{2})$ and $(k,m)=(1,0)$ (corresponding to the canonical Bessel mapping of $\phi$) and $(k,m)=(\frac{1}{2},0)$ (corresponding to the canonical tight frame of $\phi$).

It is possible to generalize Theorem 4.1 by considering more general functions than powers of $\mathsf{T}_\phi$. For instance we could take the set $\Sigma$ of real valued functions $\mathsf g$ defined on the spectrum $\sigma (\mathsf{T}_\phi )$, which are measurable with respect to the spectral measure of $\mathsf{T}_\phi$ and such that $\mathsf{g}$ and $\widetilde{\mathsf{g}}: =1/\mathsf{g}$ are bounded on compact subsets of $\sigma (\mathsf{T}_\phi )$. For every $\mathsf{g} \in \Sigma$, we denote by ${\mathcal H}_\mathsf{g}$ the Hilbert space completion of $D(\mathsf{g}(\mathsf{T}_\phi ))$ with respect to the norm $\Vert f\Vert _\mathsf{g}= \Vert \mathsf{g}(\mathsf{T}_\phi )f\Vert$, $f\in D(\mathsf{g}(\mathsf{T}_\phi ))$. As shown in [2, Sec. 10.4], we get an LHS if the order is defined by $\mathsf{h} \preceq \mathsf{g} \;\Longleftrightarrow \;\exists \;\gamma >0$, such that $\mathsf{h} \le \gamma \mathsf{g}$. We put $\mathsf{i}(t):=t, \; t\in \sigma (\mathsf{T}_\phi )$. Then, we get the following:

Theorem 4.2

Let $\phi$ be a lower semi-frame of ${\mathcal H}$ with ${\mathcal D}(C_\phi )$ dense and $\mathsf{g},\mathsf{h}$ non-negative functions from $\Sigma$, with $\mathsf{g}\succeq \mathsf{h}$. Suppose that $\widetilde{\mathsf{g}}$ and $\mathsf{h}\,\widetilde{\mathsf{g}}$ are bounded functions. Then, the following statements hold.

• (i) $\widetilde{\mathsf{g}}(\mathsf{T}_\phi )\phi$ is a Bessel mapping of ${\mathcal H}_\mathsf{h}$ if and only if $\mathsf{i}^{1/2}\mathsf{h}\preceq \mathsf{g}$;

• (ii) $\widetilde{\mathsf{g}}(\mathsf{T}_\phi )\phi$ is a lower semi-frame of ${\mathcal H}_\mathsf{h}$ if and only if $\mathsf{h} \preceq \mathsf{g}\preceq \mathsf{i}^{1/2}\mathsf{h}$;

• (iii) $\widetilde{\mathsf{g}}(\mathsf{T}_\phi )\phi$ is a frame of ${\mathcal H}_\mathsf{h}$ if and only if $\widetilde{\mathsf{g}}(\mathsf{T}_\phi )\phi$ is a Parseval frame of ${\mathcal H}_\mathsf{h}$, if and only if $\mathsf{i}^{1/2}\mathsf{h}\preceq \mathsf{g}$.

Proof

The proof is similar to that of Theorem 4.1. In fact, for $f\in {\mathcal H}_\mathsf{h}$, using the functional calculus for $\mathsf{T}_\phi$, we have:

$$\begin{aligned} \int _{X} | \langle {f} |{\widetilde{\mathsf{g}}(\mathsf{T}_\phi ) \phi _x} \rangle |_\mathsf{h}^2 \,\mathrm {d}\mu (x)= & {} \int _{X} | \langle {\mathsf{h}(\mathsf{T}_\phi )f} |{(\mathsf{h}\widetilde{\mathsf{g}})(\mathsf{T}_\phi ) \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \int _{X} | \langle {(\mathsf{h}^2\widetilde{\mathsf{g}})(\mathsf{T}_\phi ) f} |{ \phi _x} \rangle |^2\,\mathrm {d}\mu (x)\nonumber \\= & {} \Vert \mathsf{T}_\phi ^{1/2} (\mathsf{h}^2\widetilde{\mathsf{g}})(\mathsf{T}_\phi ) f \Vert ^2= \Vert (\mathsf{h}^2\widetilde{\mathsf{g}}\mathsf{i}^{1/2})(\mathsf{T}_\phi ) f \Vert ^2. \end{aligned}$$

(4.2)

Thus, for instance, if $\mathsf{i}^{1/2}\mathsf{h}\preceq \mathsf{g}$, then $\mathsf{i}^{1/2}\mathsf{h}^2\widetilde{\mathsf{g}}\preceq \mathsf{h}$; hence, $\widetilde{\mathsf{g}}(\mathsf{T}_\phi )\phi$ is a Bessel mapping of ${\mathcal H}_\mathsf{h}$. The rest of the proof is analogous. $\square$

5 Metric Operators

The generalized frame operator of a total weakly measurable function $\phi$ with $D(C_\phi )$ dense is an example of metric operator in the sense of the following definition [10].

Definition 5.1

By a metric operator in a Hilbert space ${\mathcal H}$, we mean a strictly positive self-adjoint operator G, that is, $G > 0$ or $\langle {Gf} |{f} \rangle \ge 0$ for every $f \in D(G)$ and $\langle {Gf} |{f} \rangle = 0$ if and only if $f= 0$.

Let S be an unbounded closed operator with dense domain D(S). As usual, define the graph norm of S:

$$\begin{aligned} \langle {f} |{g} \rangle _\mathrm{gr}&:= \langle {f} |{g} \rangle + \langle {Sf} |{Sg} \rangle , \; f,g \in D(S),\\ \left\| f \right\| _\mathrm{gr} ^2&= \left\| f \right\| _{} ^2 + \left\| Sf \right\| _{} ^2 . \end{aligned}$$

Then, the norm $\left\| \cdot \right\| _\mathrm{gr}$ makes D(S) into a Hilbert space continuously embedded into ${\mathcal H}$. For $f\in D(S^*S)$, we may write $\left\| f \right\| _{\mathrm{gr}} ^2 = \langle {f} |{(I+ S^*S)f} \rangle$. Note that the operator $S^*S = |S|^2$ is self-adjoint and non-negative: $|S|^2 \ge 0$. In addition, $D(|S|) = D(S)$ and $N(S^*S) = N(S)$ [28, Theor. 5.39 and 5.40].

Given S as above, the operator $R_S:= I+ S^*S$ is self-adjoint, with domain $D(S^*S)$, and $R_S\ge 1$. Hence, $R_S$ is an unbounded metric operator, with bounded inverse $R_S^{-1}=( I+ S^*S) ^{-1}$. In our previous works [10, 11, 14], we have analyzed the lattice of Hilbert spaces generated by such a metric operator. In the sequel, we summarize this discussion, keeping the same notations.

In the general case where both the metric operator G and its inverse $G^{-1}$ are unbounded, the lattice is given in Fig. 1. Given the metric operator G, equip the domain $D(G^{1/2})$ with the following norm:

$$\begin{aligned} \left\| f \right\| _{R_G} ^2 = \left\| (I+G)^{1/2}f \right\| _{} ^2,\; f \in D(G^{1/2}). \end{aligned}$$

(5.1)

Since this norm is equivalent to the graph norm of $G^{1/2}$, this makes $D(G^{1/2})$ into a Hilbert space, denoted ${\mathcal H}(R_G)$, dense in ${\mathcal H}$. Next, we equip ${\mathcal H}(R_G)$ with the norm $\left\| f \right\| _{G} ^2 := \left\| G^{1/2}f \right\| _{} ^2$ and denote by ${\mathcal H}(G)$ the completion of ${\mathcal H}(R_G)$ in that norm and corresponding inner product $\langle {\cdot } |{\cdot } \rangle _G := \langle {G^{1/2}\cdot } |{G^{1/2}\cdot } \rangle$. Hence, we have ${\mathcal H}(R_G) = {\mathcal H}\cap {\mathcal H}(G)$, with the so-called projective norm [5, Sec. I.2.1], which here is simply the graph norm of $G^{1/2}$.

Next, we proceed in the same way with the inverse operator $G^{-1}$, and we obtain another Hilbert space, ${\mathcal H}(G^{-1})$. Then, we consider the lattice generated by ${\mathcal H}(G)$ and ${\mathcal H}(G^{-1})$ with the operations:

$$\begin{aligned} {\mathcal H}_1 \wedge {\mathcal H}_2:= & {} {\mathcal H}_1 \cap {\mathcal H}_2 \, , \end{aligned}$$

(5.2)

$$\begin{aligned} {\mathcal H}_1 \vee {\mathcal H}_2:= & {} {\mathcal H}_1 + {\mathcal H}_2 \, , \end{aligned}$$

(5.3)

as shown in Fig. 1. Here, every embedding, denoted by an arrow, is continuous and has dense range. Taking conjugate duals, it is easy to see that one has:

$$\begin{aligned} {\mathcal H}(R_G)^\times= & {} {\mathcal H}(R_G^{-1}) = {\mathcal H}+ {\mathcal H}(G^{-1}), \end{aligned}$$

(5.4)

$$\begin{aligned} {\mathcal H}(R_{G^{-1}})^\times= & {} {\mathcal H}(R_{G^{-1}}^{-1}) = {\mathcal H}+ {\mathcal H}(G). \end{aligned}$$

(5.5)

In these relations, the r.h.s. is meant to carry the inductive norm (and topology) [5, Sec. I.2.1], so that both sides are in fact unitary equivalent, hence identified.

Fig. 1

The lattice of Hilbert spaces generated by a metric operator

At this stage, we return to the construction in terms of the closed unbounded operator S. We have to envisage two cases.

5.1 (i) An Unbounded Metric Operator

We take as metric operator $G_1= I+ S^*S$, which is unbounded, with $G_1 > 1$ and bounded inverse. Then, the norm $\left\| \cdot \right\| _{G_1}$ is equivalent to the norm $\left\| \cdot \right\| _{R_{G_1}}$ on $D(G_1^{1/2}) = D(S)$, so that ${\mathcal H}(G_1) = {\mathcal H}(R_{G_1})$ as vector spaces and thus also ${\mathcal H}(G_1^{-1}) = {\mathcal H}(R_{G_1}^{-1})$. On the other hand, $G_1^{-1}$ is bounded. Hence, we get the triplet:

$$\begin{aligned} {\mathcal H}(G_1) \; \subset \; {\mathcal H}\; \subset \; {\mathcal H}(G_1^{-1}) = {\mathcal H}({G_1})^\times . \end{aligned}$$

(5.6)

Next, following Example 3 of [21], if we take an ONB $\{e_n\}$ of $D(G_1^{1/2}) =D(S)$, contained in $D(S^*S) ={\mathcal H}({G_1} )$, then $\{G e_n\} = \{(1= I+ S^*S)e_n\}$ is a lower semi-frame of ${\mathcal H}$.

Actually, the triplet (5.6) is the central part of the discrete scale of Hilbert spaces $V_{{\mathcal G}}$ built on the powers of ${G_1^{1/2}}$. This means that $V_{{\mathcal G}_1}:= \{{\mathcal H}_{n}, n \in \mathbb {Z}\}$, where ${\mathcal H}_{n} = D(G_1 ^{n/2}), n\in \mathbb {N}$, with a norm equivalent to the graph norm, and ${\mathcal H}_{-n} ={\mathcal H}_{n}^\times$:

$$\begin{aligned} \cdots \subset \; {\mathcal H}_{2}\; \subset \;{\mathcal H}_{1}\; \subset \; {\mathcal H}\; \subset \; {\mathcal H}_{-1} \subset \; {\mathcal H}_{-2} \subset \; \cdots \end{aligned}$$

(5.7)

Thus, ${\mathcal H}_{1} = {\mathcal H}({G_1}^{1/2} ) = D(S)$, ${\mathcal H}_{2} = {\mathcal H}({G_1} ) = D(S^*S)$, and ${\mathcal H}_{-1} = {\mathcal H}({G_1}^{-1})$. Then, $G_1^{1/2}$ is a unitary operator from ${\mathcal H}_1$ onto ${\mathcal H}$ and, more generally, from ${\mathcal H}_n$ onto ${\mathcal H}_{n-1}$. In the same way, $G_1$ is a unitary operator from ${\mathcal H}_n$ onto ${\mathcal H}_{n-2}$ and $G_1^{-1}$ is a unitary operator from ${\mathcal H}_n$ onto ${\mathcal H}_{n+2}$.

Moreover, one may add the end spaces of the scale, namely:

$$\begin{aligned} {\mathcal H}_{\infty }({G_1} ):=\bigcap _{n\in \mathbb {Z}} {\mathcal H}_n, \qquad {\mathcal H}_{-\infty }({G_1} ):=\bigcup _{n\in \mathbb {Z}} {\mathcal H}_{n}. \end{aligned}$$

(5.8)

In this way, we get a genuine Rigged Hilbert Space:

$$\begin{aligned} {\mathcal H}_{\infty }(G_1 ) \subset {\mathcal H}\subset {\mathcal H}_{-\infty }({G_1} ). \end{aligned}$$

(5.9)

In fact, one can go one more step. Namely, following [5, Sec. 5.1.2], we can use quadratic interpolation theory [20] and build a continuous scale of Hilbert spaces ${\mathcal H}_{\alpha }, 0\le \alpha \le 1$, between ${\mathcal H}_{1}$ and ${\mathcal H}$, where ${\mathcal H}_{\alpha }= D(G_1^{\alpha /2})$, with the graph norm $\Vert \xi \Vert _{\alpha }^2 = \Vert \xi \Vert ^2 + \Vert G_1^{\alpha /2}\xi \Vert ^2$ or, equivalently, the norm $\left\| (I+G_1)^{\alpha /2}\xi \right\| _{} ^2$. Indeed every $G_1^\alpha , \alpha \ge 0$, is an unbounded metric operator.

Next, we define ${\mathcal H}_{-\alpha } ={\mathcal H}_{\alpha }^\times$ and iterate the construction to the full continuous scale $V_{\widetilde{\mathcal G}_1}:= \{{\mathcal H}_{\alpha }, \alpha \in \mathbb {R}\}$. Then, of course, one can replace $\mathbb {Z}$ by $\mathbb {R}$ in the definition (5.8) of the end spaces of the scale.

In the general case, $R_{G_1} = I+G_1 > 1$ is also an unbounded metric operator. Thus, we have:

$$\begin{aligned} {\mathcal H}(R_{G_1}) \; \subset \; {\mathcal H}\; \subset \; {\mathcal H}(R_{G_1}^{-1}) = {\mathcal H}(R_{G_1})^\times , \end{aligned}$$

(5.10)

and we get another Hilbert–Gel’fand triplet. Then, one can repeat the construction and obtain the Hilbert scale built on the powers of $R_{G_1}^{1/2}$, as well as other lower semi-frames of ${\mathcal H}$.

Now, if S is injective, i.e., $N( S)= \{0\},$ then $|S|^2>0$ is also an unbounded metric operator. Since $|S|>0, R_{S} = I+|S|^2 > 1$ and it is another unbounded metric operator, with bounded inverse $R_{S}^{-1}$. In both cases, one may build the corresponding Hilbert scale corresponding to the powers of S or $R_S^{1/2}$.

At this stage, we have recovered the formalism based on metric operators that we have developed for the theory of quasi-Hermitian operators, in particular non-self-adjoint Hamiltonians, as encountered in the so-called ${\mathcal P}{\mathcal T}$-symmetric quantum mechanics. We refer to [10, 11, 14] for a complete treatment. However, the case of an unbounded metric operator does not lead to many results, unless one considers a quasi-Hermitian operator [9, Def.3.1].

5.2 (ii) A Bounded Metric Operator

We take as metric operator $G_2= (I+ S^*S)^{-1}$, which is bounded, with unbounded inverse.

Since $G_2$ is bounded, things simplify, because now $D(G_2) = {\mathcal H}$. Similarly, one gets ${\mathcal H}(R_{G_2^{-1}}) = {\mathcal H}(G_2^{-1})$ and ${\mathcal H}(R_{G_2^{-1}}^{-1}) = {\mathcal H}(G_2)$. Therefore, we are left with the triplet

$$\begin{aligned} {\mathcal H}(G_2^{-1}) \;\subset \; {\mathcal H}\;\subset \; {\mathcal H}(G_2). \end{aligned}$$

(5.11)

Then, $G_2^{1/2}$ is a unitary operator from ${\mathcal H}(G_2)$ onto ${\mathcal H}$ and from ${\mathcal H}$ onto ${\mathcal H}(G_2^{-1})$, whereas $G_2^{-1/2}$ is a unitary operator ${\mathcal H}(G_2^{-1})$ onto ${\mathcal H}$ and from ${\mathcal H}$ onto ${\mathcal H}(G_2)$.

5.3 (iii) A Bounded Metric Operator with Bounded Inverse

There is a third case, which is almost trivial. If the operator S is bounded, $G_1= I+ S^*S$ and $G_2= (I+ S^*S)^{-1}$ are both bounded metric operators, with bounded inverse. Then, all nine Hilbert spaces in the lattice of Fig. 1 coincide as vector spaces, with equivalent, but different, norms.

The advantage of this situation is that it leads to strong results on the similarity of two operators. As mentioned in [8, Sec. 3], up to unitary equivalence, one may always consider that the intertwining operator defining the similarity is in fact a metric operator. Let us briefly recall these notions.

Let ${\mathcal H}, {\mathcal K}$ be Hilbert spaces, and D(A) and D(B) be dense subspaces of ${\mathcal H}$ and ${\mathcal K}$, respectively, $A:D(A) \rightarrow {\mathcal H}$, $B: D(B) \rightarrow {\mathcal K}$ two linear operators. A (metric) bounded operator $T:{\mathcal H}\rightarrow {\mathcal K}$ is called a intertwining operator for A and B if:

• (i) $T:D(A)\rightarrow D(B)$;

• (ii) $BT\xi = TA\xi , \; \forall \, \xi \in D(A)$.

We say that A and B are similar, and write $A\sim B$, if there exists an intertwining operator T for A and B with bounded inverse $T^{-1}:{\mathcal K}\rightarrow {\mathcal H}$, intertwining for B and A.

A parallel definition (quasi-similarity) may be given in case the inverse $T^{-1}$ of the intertwining operator T is not bounded.

From the relation $A\sim B$, there follow many interesting results about the respective spectra of A and B, as described in detail in [8].

6 Transforming Functions into Frames by Metric Operators

This section concerns the second main aim of our paper. By Proposition 3.4 and Theorem 4.1, we get the following result: if $\phi$ is a lower semi-frame of ${\mathcal H}$ and $D(C_\phi )$ is dense, then $\mathsf{T}_\phi ^{-1/2}:{\mathcal H}\rightarrow {\mathcal H}$ and $\psi = \mathsf{T}_\phi ^{-1/2} \phi$ is a Parseval frame for ${\mathcal H}_\phi = {\mathcal H}$. As we already remarked, $\mathsf{T}_\phi ^{-1/2}:{\mathcal H}\rightarrow {\mathcal H}$ is a metric operator. We now want to relax the condition for $\phi$ of being a lower semi-frame. Thus, we ask the following question: for which weakly measurable functions $\phi :X\rightarrow {\mathcal H}$ does there exist a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame?³

The motivation of this question is that, in general, one tries to pass from a less regular situation to a more regular, possibly in a smaller space.

In the next result, we find some necessary or sufficient conditions for an answer to our question. As we are going to see, the recourse of the generalized frame operator is again useful.

Theorem 6.1

Let $\phi :X\rightarrow {\mathcal H}$ be a weakly measurable function with generalized frame operator $\mathsf{T}_\phi$. The following statements hold.

• (i) If $\phi$ is not total, then there does not exist a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame and $G^{-1}$ is bounded.

• (ii) If $\phi$ is a Bessel mapping and not total, then there does not exist a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame.

• (iii) If $D(C_\phi )$ is not dense, then there does not exist a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame.

• (iv) If $D(C_\phi )$ is dense and $\phi$ is a lower semi-frame, then there exists a metric operator G on ${\mathcal H}$, such that $G\phi$ is a Parseval frame. In particular, a possible choice is $G=\mathsf{T}_\phi ^{-1/2}$.

• (v) If $\phi$ is total, $D(C_\phi )$ is dense and $\phi _x\in R(\mathsf{T}_\phi )$ for all $x\in X$, then there exists a metric operator G on ${\mathcal H}$, such that $G\phi$ is a Parseval frame. In particular, a possible choice is $G=\mathsf{T}_\phi ^{-1/2}$.

Proof

• (i) Suppose that there exists a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame and $G^{-1}$ is bounded. By hypothesis, there exists $f\in {\mathcal H}, f\ne 0$, such that $\langle {\phi _x} |{f} \rangle =0$ for a.e. $x\in X$. Since G is self-adjoint and $G^{-1}$ is bounded, then $G^{-1}$ is defined on ${\mathcal H}$. Hence $\langle {G\phi _x} |{G^{-1}f} \rangle = \langle {\phi _x} |{GG^{-1}f} \rangle =0$ for a.e. $x\in X$, which implies that $G^{-1}f=0$, i.e., the contradiction $f=0$.

• (ii) Suppose that there exists a metric operator G, such that $G\phi$ is a frame, and let $\mathsf{m}$ be a lower bound of $G\phi$. Then, for all $f\in D(G)$:

  $$\begin{aligned} \mathsf{m}\Vert f\Vert ^2\le \int _X | \langle {f} |{G\phi _x} \rangle |^2d\mu (x)=\int _X | \langle {Gf} |{\phi _x} \rangle |^2d\mu (x)\le \mathsf{M}\Vert Gf\Vert ^2, \end{aligned}$$

  (6.1)

  where $\mathsf{M}$ is an upper bound of $\phi$. Thus, (6.1) implies that $G^{-1}$ is bounded. By the previous point, we get then a contradiction.

• (iii) Suppose that there exists a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame. Then, $\int _X | \langle {f} |{G\phi _x} \rangle |^2 d\mu (x)<\infty$ for all $f\in {\mathcal H}.$ In particular, for $f\in D(G)$, we have:

  $$\begin{aligned} \int _X | \langle {Gf} |{\phi _x} \rangle |^2 d\mu (x)<\infty , \end{aligned}$$

  so $R(G)\subseteq D(C_\phi )$. This means that $N(G)\supseteq D(C_\phi )^\perp \ne \{0\}$ which contradicts the property that G is a metric operator.

• (iv) This follows by Proposition 3.4.

• (v) The statement can be proved with a similar argument to that of Proposition 3.4 (Proposition 3.2), and thus, we give only a sketch of the proof. First of all, we note that $\mathsf{T}_\phi$ has domain dense in ${\mathcal H}$, $D(\mathsf{T}_\phi )\subseteq D(\mathsf{T}_\phi ^{1/2})$ and also that $\mathsf{T}_\phi$ is injective ($\phi$ is total). Let $f\in D(\mathsf{T}_\phi )$, and then:

  $$\begin{aligned} \int _{X} | \langle {f} |{\mathsf{T}_\phi ^{-1/2} \phi _x} \rangle |^2 \,\mathrm {d}\mu (x) = \Vert \mathsf{T}_\phi ^{1/2} \mathsf{T}_\phi ^{-1/2} f \Vert ^2= \Vert f \Vert ^2. \end{aligned}$$

  Now, a standard argument of density concludes that $\psi$ is a Parseval frame of ${\mathcal H}$. $\square$

Example 6.2

Let $\{e_n\}_{n\in \mathbb {N}}$ be an orthonormal basis, $\phi =\{e_1+e_n\}_{n\ge 2}$ and $\psi =\{e_n\}_{n\ge 2}$. Both $\phi$ and $\psi$ cannot be transformed into frames of ${\mathcal H}$ by a metric operator. Indeed, $D(C_\phi )=\{e_1\}^\perp$ and $\psi$ is a Bessel sequence but not total.

We will consider more examples (concerning, in particular, lower semi-frames) in the next section. As an immediate consequence of Theorem 6.1, we get the following result (compare with [25, Corollary II.1]). We recall that two sequences $\{\phi _n\}_{n\in \mathbb {N}},\{\psi _n\}_{n\in \mathbb {N}}$ are said bi-orthogonal if $\langle {\psi _n} |{\phi _m} \rangle =\delta _{n,m}$, the Kronecker symbol.

Corollary 6.3

Let $\phi :=\{\phi _n\}_{n\in \mathbb {N}},\psi :=\{\psi _n\}_{n\in \mathbb {N}}$ be bi-orthogonal and total sequences of ${\mathcal H}$. Then, $D(C_\phi )$ is dense. Moreover, if $\mathsf{T}_\phi$ is the generalized frame operator of $\phi$, then $\phi _n\in R(\mathsf{T}_\phi )\subset R(\mathsf{T}_\phi ^{1/2})$ for all $n\in \mathbb {N}$ and $\{\mathsf{T}_\phi ^{-1/2}\phi _n\}_{n\in \mathbb {N}}$ is an orthonormal basis of ${\mathcal H}$.

Proof

Then, $D(C_\phi )$ is dense, because it contains the total sequence $\psi$, and moreover, $\psi _n\in D(T_\phi )$ and $\mathsf{T}_\phi \psi _n=\phi _n$ for all $n\in \mathbb {N}$ (which also gives $\phi _n\in R(\mathsf{T}_\phi )\subset R(\mathsf{T}_\phi ^{1/2})$). Now, by Theorem 6.1(iv), $\{\mathsf{T}_\phi ^{-1/2}\phi _n\}_{n\in \mathbb {N}}$ is a Parseval frame of ${\mathcal H}$, but since $\langle {\mathsf{T}_\phi ^{-1/2}\phi _n} |{\mathsf{T}_\phi ^{-1/2}\phi _m} \rangle = \langle {\psi _n} |{\phi _m} \rangle =\delta _{n,m}$, we conclude that $\{\mathsf{T}_\phi ^{-1/2}\phi _n\}_{n\in \mathbb {N}}$ is in particular an orthonormal basis of ${\mathcal H}$. $\square$

The problem about transforming functions in frames is still open. However, in the light of Theorem 6.1, one may formulate a new version of the problem: given a weakly measurable function $\phi :X\rightarrow {\mathcal H}$, is it true that there exists a metric operator G on ${\mathcal H}$, such that $G\phi$ is a frame if and only if $\phi$ is total and $D(C_\phi )$ is dense?

7 Examples

In this final section, we exhibit several examples of lower semi-frames, mostly taken from our previous works.

7.1 (1) Sequences of Exponential Functions

Let ${\mathcal H}=L^2(0,1)$ and $g\in L^2(0,1)$. A weighted exponential sequence of g is $\mathcal {E}(g,b):=\{g_n\}_{n\in \mathbb {Z}}=\{g(x)e^{2\pi i n b x}\}_{n\in \mathbb {Z}}$ with $b>0$. In [22, Remark 6.6], it was proved that if $0<b\le 1$, then $\mathcal {E}(g,b)$ is a lower semi-frame if and only if g is bounded away from zero, i.e., $|g(x)|\ge A$ for some $A>0$ and a.e. $x\in (0,1)$. Moreover, by [22, Corollary 6.5], the analysis operator of $\mathcal {E}(g,b)$ is densely defined and the generalized frame operator $\mathsf{T}_g$ of $\mathcal {E}(g,b)$ is the multiplication operator by $\frac{1}{b}|g|^2$, that is:

$$\begin{aligned} D(\mathsf{T}_g)=\{f\in L^2(0,1): f|g|^2\in L^2(0,1)\} \quad \text {and}\quad \mathsf{T}_g f=\frac{1}{b}|g|^2 f, \quad \forall f\in D(\mathsf{T}_g). \end{aligned}$$

Thus, for $k\ge 0$ and $n\in \mathbb {Z}$, $(\mathsf{T}_g^{-k}g_n)(x)=g(x)/|g(x)|^{-2k}e^{2\pi i n b x},$ i.e., $\mathsf{T}_g^{-k}\mathcal {E}(g,$$b)=\mathcal {E}(g/|g|^{-2k},b)$. It is not difficult to check that the conclusions of Theorem 4.1 hold.

7.2 (2) A Reproducing Kernel Hilbert space

Following [13], we consider a reproducing kernel Hilbert space of (nice) functions on a measure space $(X, \mu )$, with kernel function $k_x, x\in X$; that is, $f(x)= \langle {f} |{k_x} \rangle _K,\, \forall f\in {\mathcal H}_{K}$. Choose the weight function $m(x) >1$ and define the Hilbert scale ${\mathcal H}_k, \, k\in \mathbb {Z}$, determined by the multiplication operator $Af(x) = m(x) f(x), \, \forall x\in X$. Thus, we have, for $n \ge 1$ $({\overline{n}} = -n):$

$$\begin{aligned} {\mathcal H}_{2n} \subset {\mathcal H}_{n} \subset {\mathcal H}_K \subset {\mathcal H}_{\overline{n}} \subset {\mathcal H}_{2\overline{n}}. \end{aligned}$$

(7.1)

Then, define the measurable functions $\phi _x = k_x m^n(x), \psi _x = k_x m^{-n}(x)$, so that $C_\psi : {\mathcal H}_K \rightarrow {\mathcal H}_n, \, C_\phi : {\mathcal H}_K \rightarrow {\mathcal H}_{\overline{n}}$ continuously, and they are dual of each other. One has indeed $\langle {\phi _x} |{g} \rangle _K = \overline{g(x)}\,m^n(x) \in {\mathcal H}_{\overline{n}}$ and $\langle {\psi _x} |{g} \rangle _K = \overline{g(x)}\,m^{-n}(x) \in {\mathcal H}_{n}$, which implies duality. Thus, $(\psi , \phi )$ is a reproducing pair with $S_{\psi , \phi }=I$, where $\psi$ is an upper semi-frame and $\phi$ a lower semi-frame.

Now, we concentrate on the lower semi-frame $\phi$. First, we have $D(C_\phi ) = {\mathcal H}_{n}$, which is dense, so that ${\mathcal H}_\phi = {\mathcal H}$. Let us compute the sesquilinear form:

$$\begin{aligned} \Omega _\phi (f,g)= & {} { \langle { {C}_\phi f} |{{C}_\phi g} \rangle }_K = \int _X f(x) m(x)^n\,\overline{g(x)}m(x)^n \,\mathrm {d}\mu (x),\quad f,g\in D(C_\phi )\\= & {} \langle {\mathsf{T}_\phi ^{1/ 2} f} |{\mathsf{T}_\phi ^{1/2} g} \rangle _K. \end{aligned}$$

Therefore, we have $(\mathsf{T}^{1/ 2} _\phi f)(x) = f(x)\, m(x)^{n}$, and therefore, $(\mathsf{T}_\phi f)(x) = f(x)\, m(x)^{2n}$. Hence, $D(\mathsf{T}_\phi ) = {\mathcal H}_{2n} \subset D(C_\phi ) = {\mathcal H}_{n}$ and $D(\mathsf{T}^{1/ 2}_\phi ) = {\mathcal H}_{n}$, see (7.1) above.

Next, $\mathsf{T}^{-1/ 2}_\phi$ is the operator of multiplication by $m^{-n}$, and indeed, it is bounded in ${\mathcal H}$. Finally, since $\phi$ is a lower semi-frame, $\chi := \mathsf{T}^{-1/ 2}_\phi \phi$ is a frame in ${\mathcal H}$, by Proposition 3.2.

7.3 (3) Wavelets on the Sphere

The continuous wavelet transform on the 2-sphere $\mathbb {S}^2$ has been analyzed in [4]. For an axisymmetric (zonal) mother wavelet $\phi \in {\mathcal H}=L^2(\mathbb {S}^2,\,\mathrm {d}\mu )$, define the family of spherical wavelets:

$$\begin{aligned} \phi _{\varrho ,a}:=R_\varrho D_a\phi , \text{ where } (\varrho ,a)\in X:= SO(3)\times \mathbb {R}^+. \end{aligned}$$

Here, $D_a$ denotes the stereographic dilation operator and $R_\rho$ the unitary rotation on $\mathbb {S}^2$.

Given the wavelet $\phi$, it is known that the operator $S_\phi$ is diagonal in Fourier space (harmonic analysis on the 2-sphere reduces to expansions in spherical harmonics $Y_l^m, \, l \in \mathbb {N}_0, m= -l, \ldots , l$), and thus, it is given by a Fourier multiplier $\widehat{S_\phi f}(l,n)=s_\phi (l)\widehat{f}(l,n)$ with the symbol $s_\phi$ given by:

$$\begin{aligned} s_\phi (l):=\frac{8\pi ^2}{2l+1}\sum _{|m|\le l}\int _0^\infty \big |\widehat{D_a\phi }(l,m)\big |^2\frac{\,\mathrm {d}a}{a^3},\quad l\in \mathbb {N}_0, \end{aligned}$$

where $\widehat{D_a \phi }(l,m):= \langle {Y_l^m } |{D_a \phi } \rangle$ is the Fourier coefficient of $D_a \phi$.

If one has $\mathsf{d}\le s_\phi (l) \le \mathsf{c}$, for every $l \in \mathbb {N}$, then the wavelet $\phi$ is admissible and a frame in $L^2(\mathbb {S}^2,\,\mathrm {d}\mu )$. However, it has been shown in [29] that the reconstruction formula converges under the weaker condition $\mathsf{d}\le s_\phi (l)<\infty$ for all $l\in \mathbb {N}_0$. In that case, $\phi$ is not admissible and is a lower semi-frame, with $S_\phi$ unbounded and densely defined. The domain of $S_\phi$ is the following:

$$\begin{aligned} D(S_\phi )= & {} \{f \in L^2(\mathbb {S}^2,\,\mathrm {d}\mu ) : |\widehat{f}(l,n)|\\\le & {} \frac{1}{ s_\phi (l)} |\widehat{h}(l,n)| , \text { for some } h\in L^2(\mathbb {S}^2,\,\mathrm {d}\mu )\}. \end{aligned}$$

This domain contains, in particular, the set of band-limited functions, i.e., functions f, such that $\widehat{f}(l,n) = 0, \; \forall \,l \ge N_1$, for some $N_1 < \infty$, which is dense in $L^2$. Since $S_\phi ={D}_{\phi } {C}_{\phi }$, it follows that $D(S_\phi ) \subset D(C_\phi )$, and hence, $D(C_\phi )$ is dense as well and ${\mathcal H}_\phi = {\mathcal H}$.

We proceed as in Case (2) and consider the sesquilinear form:

$$\begin{aligned} \Omega _\phi (f,g) = { \langle { {C}_\phi f} |{{C}_\phi g} \rangle }= \langle {\mathsf{T}_\phi ^{1/ 2} f} |{\mathsf{T}_\phi ^{1/2} g} \rangle , \quad f,g\in D(C_\phi ). \end{aligned}$$

Since $D(C_\phi )$ is dense, we get $\mathsf{T}_\phi :=|C_\phi |^2=C_\phi ^* C_\phi$, $\mathsf{T}_\phi ^{1/2} = |C_\phi |$ and $D(\mathsf{T}_\phi ) \subset D(\mathsf{T}_\phi ^{1/2}) = D(C_\phi ).$

Finally, since $\phi$ is a lower semi-frame, $\chi := \mathsf{T}^{-1/ 2}_\phi \phi$ is a frame in ${\mathcal H}$, by Proposition 3.2, as before.