Lower Semi-frames, Frames, and Metric Operators

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.


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, μ) with a Radon measure μ. A function φ : X → H, x → φ x is said to be weakly measurable if for every f ∈ H the function x → f |φ x is measurable. A weakly measurable function φ is said to be μ-total if f |φ x = 0 for a.e. x ∈ X implies that f = 0. A weakly measurable function φ is a continuous frame of H if there exist constants 0 < m ≤ M < ∞ (the frame bounds), such that: , is called the analysis operator of φ. We can define the sesquilinear form: and associate a positive self-adjoint operator T φ in the space H φ , the closure of D(C φ ) in H, by Kato's representation theorem [26], which we call generalized frame operator. When φ is a lower semi-frame of H, then the range of T φ is H φ and the function ψ : X → H, defined by ψ x = T −1 φ P φ φ x , x ∈ X, where P φ is the orthogonal projection onto H φ , is a Bessel mapping and the reconstruction formula: holds for every f ∈ D(C φ ) 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 φ : X → H with D(C φ ) dense in H, we consider general powers T −k φ φ with k ≥ 0. These functions are Bessel mappings, frames or lower semi-frames in the space H(T m φ ) (given by the domain of T m φ and the inner product T m φ ·|T m φ · ) with m ≥ 0 according to a simple relation between k and m (see Theorem 4.1).
When φ : X → H is a μ-total weakly measurable function with D(C φ ) dense, then T φ is in particular a metric operator, i.e., a strictly positive selfadjoint operator. Metric operators are a topic familiar in the theory of the so-called PT -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 φ : X → H is a lower semi-frame with D(C φ ) dense, then T −1/2 φ φ is a Parseval frame of H. This inspired us to consider the following more general problem.
Question for which weakly measurable functions φ : X → H, there exists a metric operator G on H, such that φ x ∈ D(G) for all x ∈ X and Gφ is a frame? Partial answers to this problem are given in Theorem 6.1. In particular, necessary conditions are that D(C φ ) is dense, and that φ is μ-total if φ is in addition a Bessel mapping. In the discrete case, if φ : N → H is a Schauder basis, then the problem has a positive solution (more precisely, one can again take G = T −1/2 φ and T −1/2 φ φ 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 T φ , whose properties are more convenient that those of the standard frame operator S φ . 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.

Preliminaries
Before proceeding, we list further definitions and conventions. The framework is a (separable) Hilbert space H, with the inner product ·|· linear in the first factor. GL(H) denotes the set of all invertible bounded operators on H with bounded inverse. Throughout the paper, we will consider weakly measurable functions φ : X → H, where (X, μ) is a locally compact space with a Radon measure μ.
Given a continuous frame φ, the analysis operator is defined and bounded on H, i.e., C φ : H → L 2 (X, dμ) 1 and the corresponding synthesis operator C * φ : L 2 (X, dμ) → H is defined as (the integral being understood in the weak sense, as usual): Moreover, we set S φ := C * φ C φ , which is self-adjoint. Then, it follows that: Thus, for continuous frames, S φ and S −1 Following [6,7], we will say that a function φ is a semi-frame if it satisfies only one of the frame inequalities in (1.1). We already introduced the lower semi-frames (if φ satisfies (1.2), then it could still be a frame, and hence, we say that the lower semi-frame φ is proper if it is not a frame). Note that the lower frame inequality automatically implies that φ is μ-total. On the other hand, a weakly measurable function φ is an upper semi-frame if is μ-total, that is, N (C φ ) = {0}, and there exists M < ∞, such that: Thus, an upper semi-frame is a total Bessel mapping [24]. Notice that this definition does not forbid φ to be a frame. Thus, we say that φ is a proper upper semi-frame if it is not a frame. If φ is a proper upper semi-frame, S φ is bounded and S −1 φ is unbounded, as follows immediately from (2.2). In the lower case, however, the definition of S φ must be changed, since the domain D(C φ ) need not be dense, so that C * φ may not exist. Instead, following [6, Sec. 2], one defines the synthesis operator as: on the domain of all elements F for which the integral in (2.3) converges weakly in H, and then, S φ := D φ C φ . With this definition, it is shown in [6,Sec. 2] that if φ is a proper lower semi-frame, then S φ is unbounded and S −1 φ 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 φ, ψ are dual to each other (the relation is symmetric) if one has: (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 D 1 × D 2 : By assumption, there exists c > 0, such that: Hence, there exists a bounded operator T n , such that Ω n ψ,φ (f, g) = T n f |g . Applying the Banach-Steinhaus theorem to the functional g → Ω ψ,φ (f, g), one gets that the operator T ψ,φ associated with Ω ψ,φ is defined on the whole of H. Doing the same with T * n , as in [21,Prop. 7], we conclude that the form Ω ψ,φ is bounded on H × H.
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 Ω does not imply the supremum condition.
Then, φ : R → H is a weakly measurable function and a Bessel mapping, because: In conclusion, the sesquilinear form

The Generalized Frame Operator T φ
In the previous section, we defined the frame operator S φ for a lower semiframe. However, this operator lacks good properties, in general (for instance S φ need not be self-adjoint like in the case of an upper semi-frame, even if S φ is non-negative). In this section, we are going to construct a new operator associated with φ which plays the rôle of S φ 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 φ 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 φ ) need not be dense in H. It is useful to work with the Hilbert space H φ made of the closure of D(C φ ) endowed with the topology of H.
The analysis operator C φ is closed [6, Lemma 2.1]. Therefore, the sesquilinear form: is non-negative and closed. By Kato's second representation theorem [26,Theorem 2.23], there exists an operator T φ : 2 We use this sans serif font to avoid confusion with the generalized synthesis operator T φ introduced in our papers about reproducing pairs [12,13,15].
We call T φ the generalized frame operator of φ. The motivation behind this name is that when φ is a continuous frame, then T φ = S φ . 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 φ ) is dense, then, of course, H φ = H (i.e., T φ is a non-negative self-adjoint operator on H) and S φ ⊂ T φ . In [22], it was proved that in the discrete setting (X = N) we may have a strict inclusion S φ T φ (recall that in our paper S φ is weakly defined; therefore, in the discrete setting, it corresponds to the operator W φ in [22]). Since and |C φ | are the adjoint and the modulus of C φ when we think of it as an operator The following characterization can be proved as in [21]. (ii) Ω φ is bounded from below by m, that is: (iii) C φ is bounded from below by √ m, that is: (iv) T φ is bounded from below by m, that is: Now, assume that φ is a lower semi-frame of H. By [6, Proposition 2.6], there exists a Bessel mapping ψ : X → H dual to φ, i.e., for some M > 0: The following statements hold.
In conclusion, f |g = Cf |CT −1 g , for all f, g ∈ D(C). Therefore, if C = C φ is densely defined, the dual is found.
If D(C φ ) is not dense, we can proceed with the projection P φ as before.
Finally, as mentioned in [22] for the discrete case, calculations similar to (3.2) show the next result.
Thus, following the standard terminology in frame theory, we can call T −1 φ P φ φ and T −1/2 φ P φ φ the canonical dual Bessel mapping and the canonical tight frame of φ. Now, we exploit Proposition 5 of [21] and the discussion after it.

Proposition 3.5. Let φ be a weakly measurable function of H. Then, φ is a lower semi-frame of H if and only if there exists an inner product ·|·
is complete, continuously embedded into H, and for some α, m, M > 0, one has: Proof. It is sufficient to take f 2 Let φ be a lower semi-frame in the Hilbert space H, with domain D(C φ ), assumed to be dense. For every x ∈ X, the map f → f |φ x is a bounded linear functional on the Hilbert space H(C φ ) := D(C φ )[ · + ]. By the Riesz Lemma, there exists an element χ φ x ∈ D(C φ ), such that: By Proposition 3.5, χ φ is a frame. We can explicitly determine the element χ φ when · + is the norm Notice that, by Proposition 3.1(iv), this norm is equivalent to the graph norm of T i.e., χ φ is the canonical dual Bessel mapping of φ. Following the notation of [8], denote by H(T 1/2 φ ) the Hilbert space D(T 1/2 φ ) with the norm · 1/2 . In the same way, denote by H(C φ ) the Hilbert space D(C φ ) with inner product C φ ·|C φ · . Hence, we have proved the following result.  We can proceed in the converse direction, i.e., starting with a frame χ ∈ D(C φ ), does there exist a lower semi-frame η of H, such that χ is the frame χ η constructed from η in the way described above? The answer is formulated in the following.

The Functions Generated by a Lower Semi-frame
Throughout this section, we continue to consider a lower semi-frame φ in H, with D(C φ ) dense.
Since T −1 φ is defined on H, we can actually apply different powers of T −1 φ on φ and get the functions T −k φ φ, k ∈ [0, ∞). Hence, we can ask for the properties of T −k φ φ. Of course, the answer depends on the Hilbert space where T −k φ φ is considered. For instance, for k = 0, we have a lower semi-frame of H and, as seen in the previous section, for k = 1 2 , we have a frame for H, while for k = 1, we have a frame for H(T 1/2 φ ). 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, Having at our disposal the notion of scale of Hilbert spaces, we now come back to a lower semi-frame φ with D(C φ ) dense and to the functions T −k φ φ with k ≥ 0. For simplicity of notation, we write in a compact way the inner product of H(T m φ ) in the following way f |g m :  Hence, taking into account that 0 ∈ ρ(T φ ) and that f 2 T m φ = T m φ f 2 , (i) and (ii) follow by comparing 2m − k + 1 2 and m. We give the details for (i) as example.
Assume that T −k φ φ is a Bessel mapping of H(T m φ ), then there exists B > 0, such that for all f ∈ D(T m φ ), we have T The other implication trivially holds by the same estimate.
Finally, combining the two cases above, we obtain that T −k φ φ is a frame of H(T m φ ) if and only if k = m + 1 2 . Moreover, in this case, T −k φ φ is actually a Parseval frame, as one can see in (4.1).
It is possible to generalize Theorem 4.1 by considering more general functions than powers of T φ . For instance we could take the set Σ of real valued functions g defined on the spectrum σ(T φ ), which are measurable with respect to the spectral measure of T φ and such that g and g := 1/g are bounded on compact subsets of σ(T φ ). For every g ∈ Σ, we denote by H g the Hilbert space completion of D(g(T φ )) with respect to the norm f g = g(T φ )f , f ∈ D(g(T φ )). As shown in [2, Sec. 10.4], we get an LHS if the order is defined by h g ⇐⇒ ∃ γ > 0, such that h ≤ γg. We put i(t) := t, t ∈ σ(T φ ). Then, we get the following:  Proof. The proof is similar to that of Theorem 4.1. In fact, for f ∈ H h , using the functional calculus for T φ , we have: Thus, for instance, if i 1/2 h g, then i 1/2 h 2 g h; hence, g(T φ )φ is a Bessel mapping of H h . The rest of the proof is analogous.

Metric Operators
The generalized frame operator of a total weakly measurable function φ with D(C φ ) dense is an example of metric operator in the sense of the following definition [10]. Given S as above, the operator R S := I + S * S is self-adjoint, with domain D(S * S), and R S ≥ 1. Hence, R S is an unbounded metric operator, with bounded inverse R −1 S = (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: in that norm and corresponding inner product ·|· G := G 1/2 ·|G 1/2 · . Hence, we have H(R G ) = H∩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, H(G −1 ). Then, we consider the lattice generated by H(G) and H(G −1 ) with the operations: 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: At this stage, we return to the construction in terms of the closed unbounded operator S. We have to envisage two cases.

(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 · G1 is equivalent to the norm · RG 1

as vector spaces and thus also
). On the other hand, G −1 1 is bounded. Hence, we get the triplet: In this way, we get a genuine Rigged Hilbert Space: with the graph norm ξ 2 α = ξ 2 + G α/2 1 ξ 2 or, equivalently, the norm (I + G 1 ) α/2 ξ 2 . Indeed every G α 1 , α ≥ 0, is an unbounded metric operator. Next, we define H −α = H × α and iterate the construction to the full continuous scale V G1 := {H α , α ∈ R}. Then, of course, one can replace Z by R in the definition (5.8) of the end spaces of the scale.
In the general case, R G1 = I + G 1 > 1 is also an unbounded metric operator. Thus, we have: 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 1/2 G1 , as well as other lower semi-frames of 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 −1 S . In both cases, one may build the corresponding Hilbert scale corresponding to the powers of S or R 1/2 S . 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 PT -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].

(ii) A Bounded Metric Operator
We take as metric operator G 2 = (I + S * S) −1 , which is bounded, with unbounded inverse. Since

(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