Continuous frames for unbounded operators

Few years ago G\u{a}vru\c{t}a gave the notions of $K$-frame and atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$ in order to decompose $\mathcal{R}(K)$, the range of $K$, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator on a Hilbert space $A$ in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.


Introduction
The notion of discrete frame was introduced by Duffin and Schaefer in 1952 [17] even though it raised on the mathematical and physical scene in 1986 with the paper of I. Daubechies, A. Grossmann, Y. Meyer because of their use in wavelet analysis. In the early '90s G. Kaiser [23] and (independently) S.T. Ali, J.P. Antoine and J.P. Gazeau [1] extended this notion to the continuous case. Over the years many extensions of frames have been introduced and studied. Most of them have been considered in the discrete case because of their wide use in applications e.g. in signal processing [17]. Frames have been studied for the whole Hilbert space or for a closed subspace until 2012, when L. Gȃvruţa [20] gave the notions of K-frame and of atomic system for a bounded operator K everywhere defined on H, thus generalizing the notion of frame and that of atomic system for a subspace due to H.G. Feichtinger and T. Werther [19]. K-frames allow to write each element of R(K), the range of K, which is not a closed subspace in general, as a combination of the elements of the K-frame, which do not necessarily belong to R(K) with K ∈ B(H). K-frames have been generalized in [4] and [21] where the notion of K-g-frames was investigated and have been further generalized in 2018 to the continuous case in [2].
Let H be a Hilbert space with inner product · |· and norm · , (X, µ) a measure space where µ is a positive measure and A a densely defined operator on H. Let ψ : x ∈ X → ψ x ∈ H be a Bessel function, i.e. ψ be such that for all f ∈ H, the map x → f |ψ x is a measurable function on X and there exists a constant β > 0 such that X | f |ψ x | 2 dµ(x) ≤ β f 2 , ∀f ∈ H. Assume that for f ∈ D(A) (the domain of A) we have the decomposition 1 Af |u = X a f (x) ψ x |u dµ(x), ∀u ∈ D(A * ).
for some a f ∈ L 2 (X, µ). If A is unbounded, the function a f can not depend continuously on f , differently to what occurs when A is bounded. In order to decompose the range of a densely defined unbounded operator A as a combination of vectors in H, we need somewhat which takes on its unboundedness. In literature there are some generalizations to the continuous case of the notion of K-frame (as, e.g., c-K-g-frames in [2]), however, as far as the author knows, the case of an unbounded operator K in H has been little considered.
In [9] this problem has been addressed in the discrete case. In the present paper both the approaches introduced in [9] are extended to the continuous setting. One of the approaches involves a Bessel function ψ and the coefficient function a f depends continuously on f ∈ D(A) only in the graph topology of A, (which is stronger than the norm of H); the other one involves a non-Bessel function ψ but the coefficient function a f depends continuously on f ∈ D(A). In the latter approach, the notions of continuous weak A-frame and continuous weak atomic system for an unbounded operator A are introduced and studied.
If θ : X → H is a continuous frame for H then of course where ζ : X → H is a dual frame of θ. In contrast, if ψ is a continuous weak A-frame, then there exists a Bessel function φ : X → H such that Ah |u = X h|φ x ψ x |u dµ(x), ∀h ∈ D(A), u ∈ D(A * ) and the action of the operator A does not appear in the weak decomposition of the range of A. Still, continuous weak A-frames clearly call to mind continuous multipliers which are the object of interest of a recent literature even though unbounded multipliers, as far the authors knows, have been little looked over. For example, some initial steps toward this direction has been done, in the discrete case, in [5,6,7,8,22] where some unbounded multipliers have been defined. Therefore this paper can spure investigation in the direction of unbounded multipliers in the continuous case.
The paper is organized as follows. In Sect. 2 we recall some well known definitions and introduce the generalized frame operator S Ψ which is the operator associated to a sesquilinear form defined by means of a function ψ : x ∈ X → ψ x ∈ H. In Sect. 3 we introduce, prove the existence (under opportune hypotheses) and study the notions of continuous weak A-frame and continuous weak atomic system for a densely defined operator A in a Hilbert space H. To go into more detail, after having introduced and studied the notion of continuous weak A-frame, Subsection 3.1 is devoted to the study of frame-related operators as the analysis, synthesis and (generalized) frame operators of a continuous weak A-frame. In Subsection 3.2 the notion of continuous weak atomic system for an unbounded operator A in Hilbert space H is given. Under some hypotheses, this notion is equivalent to that of continuous weak A-frame. Moreover, given a suitable function ψ : x ∈ X → ψ x ∈ H, for every bounded operator M ∈ B(H, L 2 (X, µ)), an operator A M can be constructed in order ψ to be a continuous weak atomic system for A M . Section 4 is devoted to the second approach to the problem of decomposing the range of an unbounded operator in Hilbert space: we consider a bounded operator K from a Hilbert space J into another one H and give some results about both continuous K-frames and continuous atomic systems for K and about their frame-related operators, then in Subsection 4.1, we use them to study the case of an unbounded closed and densely defined operator A : D(A) → H viewing it as a bounded one A : H A → H, where H A is the Hilbert space obtained by giving D(A) the graph norm.

Definitions and preliminary results
Throughout the paper we will denote by H a complex Hilbert space with inner product · |· (linear in the first entry) and induced norm · , by (X, µ) a σ-finite measure space (i.e. X can be covered with at most countably many measurable, possibly disjoint, sets {X n } n∈N of finite measure), by B(H) the Banach space of bounded linear operators from H into H. For brevity we will indicate by L 2 (X, µ) the class of all µ-measurable functions f : X → C such that by identifying functions which differ only on a µ-null subset of X. i) for all f ∈ H, the map x → f |ψ x is a measurable function on X (i.e. the function ψ is weakly measurable), ii) there exist constants α, β > 0 such that The function ψ is called a Bessel function if at least the upper condition in (1) holds. If α = β = 1 then the function ψ is called a Parseval function.
The main feature of a frame, hence of a continuous frame too, is the possibility of writing each vector of a Hilbert space as a sum of a infinite linear combination of vectors in the space getting rid of rigidness of orthonormality of the vectors of a basis and of the uniqueness of the decomposition, but still maintaining numerical stability of the reconstruction and fast convergence. By a continuous frame it is possible to represent every element of the Hilbert space by a reconstruction formula: if ψ : x ∈ X → ψ x ∈ H is a continuous frame for the Hilbert space H, then any f ∈ H can be expressed as where φ : x ∈ X → φ x ∈ H is a function called dual of ψ and the integrals have to be understood in the weak sense, as usual.
2.1. Frame-related operators and sesquilinear forms. In this section we recall the definitions of the main operators linked to a ψ : x ∈ X → ψ x ∈ H and prove some results about them. We want to drive the attention of the reader on the fact that, in contrast with the discrete case where some results involve strong convergence [9], in the continuous case we can prove our results just in weak sense.
In the sequel we will briefly indicate the range {ψ x } x∈X of a function ψ : Consider the function ψ : x ∈ X → ψ x ∈ H and the set The operator C ψ : h ∈ D(C ψ ) ⊂ H → h |ψ x ∈ L 2 (X, µ) (strongly) defined, for every h ∈ D(C ψ ) and for every x ∈ X, by is called the analysis operator of the function ψ (borrowing the terminology from frame theory).

Remark 2.2.
In general the domain of C ψ is not dense, hence C * ψ is not well-defined. An example of function whose analysis operator is densely defined can be found in Example 2.8, where D(C ψ ) = D(Ψ). Moreover, a sufficient condition for D(C ψ ) to be dense in H is that ψ x ∈ D(C ψ ) for every x ∈ X (see Lemma 2.3. [3]).
The next result will be often needed in Section 3. In contrast with [3, Lemma 2.1] we do not suppose that {ψ x } is total. Proposition 2.3. Let (X, µ) be a measure space and ψ : x ∈ X → ψ x ∈ H. The analysis operator C ψ is closed.
Proof. Consider any sequence {h n } ⊂ D(C ψ ) such that h n · → h in H and C ψ h n · 2 → c in L 2 (X, µ) as n → ∞; we shall prove that h ∈ D(C ψ ) and C ψ h = c. For every x ∈ X the functionals f ∈ D(C ψ ) → f |ψ x ∈ C are continuous hence, as n → ∞, in L 2 (X, µ). By [26,Theorem 3.12] there exists a subsequence g (k) n l = h n l |ψ x → g = h |ψ x a.e. on X. Then for every x ∈ X we have that for n → ∞ If C ψ is densely defined, let us calculate its adjoint operator: let a ∈ D(C * ψ ) with and is called the synthesis operator of the function ψ where Remark 2.4. Thus, if C ψ is densely defined, then the synthesis operator C * ψ is a densely defined closed operator.
Proposition 2.5 ( [18]). The function ψ : x ∈ X → ψ x ∈ H is Bessel with bound β > 0 if and only if the synthesis operator C * ψ is linear and bounded on L 2 (X, µ) with C * ψ L 2 ,H ≤ √ β. Moreover, the analysis operator C ψ is linear and bounded on H with C ψ H, Extending to the continuous case [14], consider the set and the mapping Ψ : D(Ψ) × D(Ψ) → C defined by Ψ is clearly a nonnegative symmetric sesquilinear form which is well defined for every f, g ∈ D(Ψ) because of the Cauchy-Schwarz inequality. It is unbounded in general. Moreover, since D(Ψ) is the largest domain such that Ψ is defined on D(Ψ)×D(Ψ), it results that where C ψ is the analysis operator defined in (2). Since C ψ is a closed operator, Ψ is a closed nonnegative symmetric sesquilinear form in H (see e.g. [ with h as in (5) (h is uniquely determined because of the density of D(Ψ)). The operator S Ψ is the greatest one whose domain is contained in D(Ψ) and for which the following representation holds and comparing with (4), we obtain Definition 2.6. The operator S Ψ : D(S Ψ ) ⊂ H → H defined by (6) will be said the generalized frame operator of the function ψ : x ∈ X → ψ x ∈ H.
Given ψ : x ∈ X → ψ x ∈ H, coherently with [3], the operator S ψ : D(S ψ ) ⊂ H → H weakly defined by is called the frame operator of ψ. It is a positive operator on its domain and symmetric indeed for every f, g ∈ D(S ψ ) but non densely defined in general. If ψ is a continuous frame for H, then the frame operator S ψ is a bounded operator in H, positive, invertible with bounded inverse (see e.g. [1]).
Remark 2.7. The generalized frame operator S Ψ and the frame operator S ψ coincide on D(S ψ ) ⊂ D(S Ψ ). If in particular ψ is a continuous frame for H, then C ψ , S ψ are defined on the whole H and C * ψ on the whole L 2 (X, µ) (see also [3]) and However, in general, they are not the same operator, as the following example shows.
Example 2.8. Let X be such that µ(X) = ∞ and having a covering made up of a countable collection {X n } n∈N of disjoint measurable subspaces of X each of measure M > 0, H a separable Hilbert space and {e n } n∈N an orthonormal basis of H. Let α > 1, β > 0 and define ψ : Then because only two, three, or six addendi in the series are different from zero, depending on the value of m. Then span{ψ x } = span{ψ n } ⊂ D(Ψ). On the other hand (span{ψ n }) ⊥ ⊂ D(Ψ), hence hence D(Ψ) is dense in H. We shall prove that there exists a f ∈ D(S Ψ ) such that f / ∈ D(S ψ ). Let f ∈ H be such that f |e n = 1 n p for every n ∈ N, for a fixed p ∈ N. We want to calculate for which values of α and β such an f ∈ H is in D(S Ψ )\D(S ψ ). For f ∈ D(Ψ) it has to be For p > β + 1 2 the first series in (7) converges, the second has general term that behaves like 1 n 2(p−α+1) hence if p > α − 1 2 too, then the series converges. To be f ∈ D(S Ψ ) the functional g ∈ D(Ψ) → X f |ψ x ψ x |g dµ(x) has to be bounded. Take any g ∈ D(Ψ), then X f |ψ x ψ x |g dµ(x) = M ∞ n=1 f |ψ n ψ n |g . Let us consider the sequence of partial sums of the series ∞ n=1 f |ψ n ψ n : and i.e. if the series ∞ k=1 h |ψ k ψ k weakly converges in H, however, if h = f and 0 < 2α − 1 − p < β the norm of s k goes to infinity as k → ∞.
Proof. The sesquilinear form Ψ is nonnegative closed and densely defined (D(C ψ ) = D(Ψ)), hence the generalized frame operator S Ψ is self-adjoint. We conclude the proof by recalling that S ψ ⊂ S Ψ .
In the following sections we will use the next two lemmas.
The operator W † is called the pseudo-inverse of the operator W .

Continuous weak A-frame and continuous atomic systems for unbounded operators
In this section we introduce and study our extension to the continuous case of the notions of discrete weak A-frame and discrete weak atomic system for a densely defined operator A on a Hilbert space, given in [9].
Remark 3.2. If X = N and µ is a counting measure, a continuous weak A-frame clearly reduces to a discrete weak A-frame in the sense of [9]. Remark 3.3. Let (X, µ) be a σ-finite measure space. If A ∈ B(H), a continuous weak A-frame is a continuous A-g-frame in the sense of [2, Definition 2.1] with Λ x = f |ψ x for every f ∈ H, with x ∈ X, since C ψ is a bounded operator in that case.
Example 3.6. Let X = R and let µ be the Lebesgue measure on R 2 . Let H = L 2 (0, 1) and let I (0,1) be the identity of L 2 (0, 1). Let us consider the differentiation operator Af = −if ′ with domain H 1 (0, 1) which is a densely defined closed operator of L 2 (0, 1) (see [27,Section 1.3]). The function ψ : t ∈ R → ψ t ∈ L 2 (0, 1) with ψ t = 2πte 2πit· I (0,1) is a continuous weak A-frame for L 2 (0, 1). Indeed, as proved in [11,Example 4.2], the function θ : t ∈ R → θ t ∈ H 1 (0, 1) ⊂ L 2 (0, 1) such that θ t := e 2πit· I (0,1) is a Parseval function in L 2 (0, 1). Hence ψ = Aθ is a continuous weak A-frame for L 2 (0, 1). Proof. By hypothesis there exists α > 0 such that for every f ∈ D(A * ) The adjoint (AF ) * is well defined and F * A * = (AF ) * by [25,Theorem 13.2]. Hence, for every h ∈ D((AF ) * ) = D(F * A * ) Proposition 3.8. Let A be a self-adjoint operator and ψ : x ∈ X → ψ x ∈ D(A) ⊂ H a continuous weak A-frame for H with lower bound α, then Aψ is a continuous weak A 2 -frame for H with the same lower bound α. Moreover, if ψ : x ∈ X → ψ x ∈ n k=1 D(A k ) ⊂ H, then A n ψ is a continuous weak A n+1 -frame for H, for every fixed n ∈ N, with the same lower bound α. In particular, if ψ : x ∈ X → ψ x ∈ n∈N D(A n ) ⊂ H is a continuous weak A-frame for H with lower bound α, then A n ψ is a continuous weak A n+1 -frame for H, for every n ∈ N, with the same lower bound α.
Proof. By hypotheses A 2 is self-adjoint with dense domain D(A 2 ) ⊂ D(A) and there exists α > 0 such that for every f ∈ D(A) Hence, for every h ∈ D(A 2 ) Fix now an arbitrary n ∈ N. If ψ : x ∈ X → ψ x ∈ D(A n ) ⊂ H, then, as before, by hypotheses both A n and A n+1 are self-adjoint with dense domain D(A n+1 ) ⊂ D(A n ) ⊂ D(A) and for every h ∈ D(A n+1 ) The last sentence in the Theorem is now obvious.
Definition 3.9. Let A be a densely defined operator and ψ : x ∈ X → ψ x ∈ H, then a function φ : The weak A-dual φ of ψ is not unique, in general. , x ∈ X n , ∀n ∈ N, then one can take φ with , x ∈ X n , ∀n ∈ N.
ii) If ψ := Aζ, where ζ : x ∈ X → ζ x ∈ D(A) ⊂ H is a continuous frame for H, then one can take as φ any dual frame of {ζ x }.

Frame-related operators of continuous weak A-frames.
In this subsection we will establish some properties of the analysis, synthesis and (generalized) frame operators of a continuous weak A-frame with A a densely defined operator. A theorem of characterization for a continuous weak A-frame is also given.
Consider the sesquilinear form Ψ defined in (3), then we can prove the following Corollary 3.12. Let A be a closable and densely defined operator, ψ a continuous weak A-frame, then the synthesis operator C * ψ is closed.
Proof. By Proposition 3.11, the domain D(C ψ ) = D(Ψ) of the closed operator C ψ is dense, hence C * ψ is closed and densely defined.

Remark 3.13.
For what has been established until now, if A is closable and densely defined and ψ is a continuous weak A-frame, by (4) the sesquilinear form Ψ is a densely defined, nonnegative closed form. Then there exists the generalized frame operator S Ψ of ψ defined as in (6) and the analysis operator C ψ is closed and densely defined. Moreover, one has Corollary 3.14. Let (X, µ) be a σ-finite measure space, A a closable, densely defined operator, ψ a continuous weak A-frame for H. Then the generalized frame operator S Ψ of ψ is self-adjoint and the frame operator S ψ is closable.
Proof. By Proposition 3.11, the domain D(Ψ) is dense, hence the thesis follows by Proposition 2.9. Proof. The proof is straightforward once observed that in our hypotheses α A * f 2 ≤ C ψ f 2 2 for every f ∈ D(A * ) and some α > 0.
The following is a theorem of characterization for continuous weak A-frames.
Theorem 3.16. Let (X, µ) be a σ-finite measure space, A a closed densely defined operator and ψ : x ∈ X → ψ x ∈ H. Then the following statements are equivalent.
i) ψ is a continuous weak A-frame for H; ii) for every f ∈ D(A * ), the map x → f |ψ x is a measurable function on X and there exists a closed densely defined extension R of C * ψ , with D(R * ) ⊃ D(A * ), such that A = RM for some M ∈ B(H, L 2 (X, µ)).
x ∈ X which is a restriction of the analysis operator C ψ . Since C ψ is closed, B is closable. B is also densely defined since D(A * ) is dense. We apply Lemma 2.11 to T 1 := A and T 2 := B noting that Bf 2 2 = X | f |ψ x | 2 dµ(x).
This proves that ψ is a continuous weak A-frame.

Atomic systems for unbounded operators A and their relation with A-
frames. Now we define our generalization to the continuous case and to unbounded operators of the notion of atomic system for K, with K ∈ B(H) [20].
Definition 3.17. Let A be a densely defined operator on H. A continuous weak atomic system for A is a function ψ : x ∈ X → ψ x ∈ H such that for all f ∈ D(A * ), the map x → f |ψ x is a measurable function on X and Remark 3.18. If ψ is a continuous weak atomic system for a densely defined operator A then, for every f ∈ D(A) and for every u ∈ D(A * ) the function g u f (x) = a f (x) ψ x |u in (8) is µ-integrable. Indeed it is absolutely integrable: fix any f ∈ D(A), u ∈ D(A * ), then by Schwarz inequality where the last inequality follows from both conditions in Definition 3.17.
The next theorem guarantees the existence of continuous weak atomic systems for densely defined operators on H. Proof. Let {e n } n∈N ⊂ D(A) be an othonormal basis for H. Then, every f ∈ H can be written as f = ∞ n=1 f |e n e n . For all n ∈ N denote with ψ n = Ae n . Let {X n } x∈N be a covering of X made up of countably many measurable disjoint sets of finite measure. It is not restrictive supposing that µ(X n ) > 0 for every n ∈ N. Then we define ψ x := ψ n µ(X n ) , x ∈ X n , n ∈ N.
For every f ∈ H the map x ∈ X → f |ψ x ∈ C is measurable because it is a step function.
Moreover, for every f ∈ D(A * ) Now, for all f ∈ D(A * ), take a f as the step function defined as follows: Then, for all f ∈ D(A * ), a f ∈ L 2 (X, µ), with and for every f ∈ D(A), u ∈ D(A * ) Therefore ψ is a continuous weak atomic system for A.
The following theorem gives a characterization of continuous weak atomic systems for A and continuous weak A-frames.
Theorem 3.20. Let (X, µ) be a σ-finite measure space, ψ : x ∈ X → ψ x ∈ and A a closable densely defined operator. Then the following statements are equivalent.
i) ψ is a continuous weak atomic system for A; ii) ψ is a continuous weak A-frame; iii) X | f |ψ x | 2 dµ(x) < ∞ for every f ∈ D(A * ) and there exists a Bessel weak A-dual φ of ψ.
Proof. i) ⇒ ii) For every f ∈ D(A * ) by the density of D(A * ) we have for some γ > 0, the last two inequalities are due to the fact that ψ is a continuous weak atomic system for A. ii) ⇒ iii) Following the proof of Theorem 3.16, there exists M ∈ B(H, L 2 (X, µ)) such that A = B * M , with B : D(A * ) → L 2 (X, µ) a closable, densely defined operator which is a restriction of the analysis operator C ψ . By the Riesz representation theorem, for every x ∈ X there exists a unique vector Moreover, for h ∈ D(A), u ∈ D(A * ) = D(B) iii) ⇒ i) It suffices to take a f : . Indeed a f ∈ L 2 (X, µ) and, for some The proof of Theorem 3.20 suggests the following If M ∈ B(H, L 2 (X, µ)) and x ∈ X denote by φ x the unique vector of H such that (M h)(x) = h |φ x for every h ∈ H. Then, there exists a closed, densely defined operator A M such that ψ is a continuous weak atomic system for A M and φ : x ∈ X → φ x ∈ H is a Bessel function which is a weak A M -dual of ψ.
Proof. Let us consider the operator B : D → L 2 (X, µ) defined for every f ∈ D by (Bf )(x) = f |ψ x , ∀x ∈ X which is a restriction of the analysis operator C ψ . Since B is densely defined, then B * , the adjoint of B, is well defined. Now fix any M ∈ B(H, L 2 (X, µ)), for every h ∈ H and any x ∈ X by the Riesz representation theorem there exists a function φ : By the same calculations than in Theorem 3.20, φ is a Bessel function. Consider the closed operator E = B * M , then E * ⊃ M * B * * ⊃ M * B and define F = E * ↾D = M * B which is closable and densely defined. Then D(F * ) is dense and ∀u ∈ D = D(F ) and ∀h ∈ D(F * ) we have It suffices now to take A M = F * .
If R(A) is weakly decomposable, then R(A * ) is weakly decomposable too.
Proposition 3.22. Let A be a densely defined operator on H, ψ a continuous weak atomic system for A and φ a Bessel weak A-dual of ψ. Then, the adjoint A * of A admits a weak decomposition and Proof. Fix any u ∈ D(A * ) then, for every h ∈ D(A) Remark 3.23. In the discrete case, i.e. for X = N and µ a counting measure, albeit a strong decomposition of A is still not guaranteed in general, the adjoint A * admits a strong decomposition (see [9,Remark 3.13]) in the sense that with {φ n } a Bessel weak A-dual of the weak A-frame {ψ n } .
Remark 3.24. Contrarily to the case in which the operator is in B(H), given a closed densely defined operator A on H and a continuous weak A-frame ψ, a weak A-dual φ of ψ is not a continuous weak A * -frame, in general. For example, if A is unbounded and φ is also a Bessel function, from the inequality with α > 0, we obtain that A is bounded, a contradiction.
We conclude this section by proving that, under suitable hypotheses, we can weakly decompose the domain of A * by means of a continuous weak A-frame.
Theorem 3.25. Let A be a closed densely defined operator with R(A) = H and A † the pseudo-inverse of A. Let ψ be a continuous weak A-frame and φ a Bessel weak A-dual of ψ. Then, the function ϑ with ϑ x := (A † ) * φ x ∈ H, for every x ∈ X, is Bessel and every u ∈ D(A * ) can be weakly decomposed as follows ∀f ∈ H, u ∈ D(A * ).
Proof. By Lemma 2.10 there exists a unique pseudo-inverse Consider the adjoint (A † ) * ∈ B(H) of A † and define ϑ x := (A † ) * φ x ∈ H, for every x ∈ X. Then, for any f ∈ H, we have and for some γ > 0 since φ is Bessel and A † is bounded. Hence, ϑ : x ∈ X → ϑ x ∈ H is a Bessel function.
Remark 3.26. In the discrete case the decomposition of the domain of D(A * ) is strong [9].

Continuous atomic systems for bounded operators between different Hilbert spaces
In this section we introduce our second approach to the generalization of the notion of (discrete) atomic system for K ∈ B(H) and of K-frame in [20], to unbounded operators in a Hilbert space in the continuous framework. Since a closed densely defined operator in a Hilbert space A : D(A) → H can be seen as a bounded operator A : H A → H between two different Hilbert spaces (with H A the Hilbert space D(A)[ · A ] where · A is the graph norm), before introducing new notions, we put the main definitions and results in [2,20] for K ∈ B(H) in terms of bounded operators from a Hilbert space into another. Later, in Section 4.1, we return to the operator A : H A → H.
Let H, J be two Hilbert spaces with inner products · |· H , · |· J and induced norms · H , · J , respectively. We denote by B(J , H) the set of bounded linear operators from J into H. For any K ∈ B(J , H) we denote by K * ∈ B(H, J ) its adjoint. H). The function ψ : x ∈ X → ψ x ∈ H is a continuous atomic system for K if for all h ∈ H, the map x → h |ψ x H is a measurable function on X and Thus the function ψ = Kξ is a continuous atomic system for K, taking a f (x) := f |ϑ x J .
In the discrete case, the decomposition of R(K), the range of K, is strong [9].
We give a result of existence of a continuous atomic system for a bounded operator. Proof. With the same notation than in Theorem 3.19 we have that where the last equality is due to the Parseval identity. The thesis follows from Theorem 3.19, with slight modifications due to the fact that K ∈ B(J , H).
The constants α, β will be called frame bounds.
It is easy to see that if K ∈ B(J , H) and θ is a continuous frame for J , then Kθ is a continuous K-frame for H. Then we give the following two examples. Corollary 4.9. Let K ∈ B(H) and ψ be a continuous K-frame for H, then ψ and K n ψ are continuous K n+1 -frames for H, for every integer n ≥ 0.
Let us give a characterization of continuous atomic systems for operators in B(J , H).
Theorem 4.10. Let ψ : x ∈ X → ψ x ∈ H and K ∈ B(J , H). Then the following are equivalent.
i) ψ is a continuous atomic system for K; ii) ψ is a continuous K-frame for H; iii) ψ is a Bessel function and there exists a Bessel function φ : X → J such that Proof. The proof follows from Theorem 3.20, with suitable adjustments, recalling that if ψ is a continuous K-frame for H, then it is a Bessel function.
As in the discrete case, Definition 4.11. Let K ∈ B(J , H) and ψ : x ∈ X → ψ x ∈ H a continuous K-frame for H. A function φ : X → H as in (10) is called a K-dual of ψ.
Example 4.12. In general, a K-dual φ : x ∈ X → φ x ∈ J of a continuous K-frame ψ : x ∈ X → ψ x ∈ H is not unique. Let us see some examples.
i) If ψ = ζ, where ζ : X → H is a continuous frame for H, then one can take φ = K * ξ : X → J where ξ : x ∈ X → ξ x ∈ H is any dual frame of ζ. ii) If ψ = Kζ, where ζ : x ∈ X → ζ x ∈ J is a continuous frame for J , then one can take as φ any dual frame of ζ.
Remark 4.13. Once at hand a continuous atomic system ψ for K, a Bessel K-dual φ : X → J as in Theorem 4.10 is a continuous atomic system for K * . Indeed, We apply Theorem 4.10 to K * and φ to conclude that φ is a continuous atomic system for K * .
Following H.G. Feichtinger and T. Werther [19], Definition 4.14. Let ψ : x ∈ X → ψ x ∈ H be a Bessel function and H 0 a closed subspace of H. The function ψ is called a continuous family of local atoms for H 0 if there exists a family of linear functionals {c x } with c x : H → C for every x ∈ X, such that We will say that the pair {ψ x , c x } provides an atomic decomposition for H 0 and γ will be called an atomic bound of {ψ x }.
If now K = P H 0 ∈ B(H) is the orthogonal projection on H 0 (P H 0 = P 2 H 0 = P * H 0 ), a continuous P H 0 -frame is a family of continuous local atoms for H 0 , similarly to [20,Theorem 5]. ii) ψ is a continuous atomic system for P H 0 ; iii) there exists α > 0 such that α P H 0 f 2 ≤ X | f |ψ x | 2 dµ(x), f ∈ H; iv) there exists a Bessel function φ : x ∈ X → φ x ∈ H such that for any f, h ∈ H.
Not even if J = H a Bessel function ψ : X → H and a K-dual φ : X → H of its are interchangeable, in general. However, if we strenghten hypotheses on K, it can be proved the existence of a function with range in H which is interchangeable with ψ in the weak decomposition of R(K) ⊂ H (see also [2,Theorem 3.2]). Theorem 4.16. Let K ∈ B(J , H) with closed range R(K). Let ψ be a continuous K-frame and φ a Bessel K-dual of its. Then, i) the function ϑ : x ∈ X → ϑ x ∈ H with ϑ x := (K † ↾R(K) ) * φ x ∈ H, for every x ∈ X, is Bessel for R(K) and interchangeable with ψ for any h ∈ R(K), i.e.
ii) ϑ is a continuous K-frame for H and K * ϑ and K * ψ are Bessel K-duals of ψ and of ϑ respectively. In particular, for every h ∈ H Proof. i) See [2, Theorem 3.2] with obvious adjustments. ii) Clearly (11) follows from i). The function ϑ is a continuous K-frame for H by i) and (11), taking for all f ∈ J , a f (x) = f |K * ψ x J , for every x ∈ X. The functions K * ϑ and K * ψ are Bessel for J , indeed for all f ∈ J , the maps x → K † ↾R(K) Kf |φ x J = f |K * ϑ x J and x → Kf |ψ x H = f |K * ψ x J are 4.1. Continuous atomic systems for unbounded operators A and continuous A-frames. The results of Section 4 can be used to generalize continuous frames for bounded operators to the case of an unbounded closed and densely defined operator A : D(A) → H viewing it as a bounded operator between two different Hilbert spaces, more precisely, from the Hilbert space H A = D(A)[ · A ] (where · A is the graph norm induced by the graph inner product · |· A ) into H.
In order to simplify notations, we come back to denote again by · |· and · the inner product and the norm of H, respectively.
We will indicate by A ♯ : H → H A the adjoint of the bounded operator A : H A → H. With this convention, if A ∈ B(H A , H), a function ψ : x ∈ X → ψ x ∈ H such that for all f ∈ H, the map x → f |ψ x is a measurable function on X is said to be i) a continuous atomic system for A if ψ is a Bessel function and there exists γ > 0 such that for all f ∈ D(A) there exists a f ∈ L 2 (X, µ), with a f 2 = X |a f (x)| 2 dµ(x) 1/2 ≤ γ f A and for every g ∈ H ii) a continuous A-frame if there exist α, β > 0 such that for every h ∈ H Theorem 4.10 and Theorem 4.18 can be summarized and rewritten as follows.
Corollary 4.22. Let ψ : x ∈ X → ψ x ∈ H and suppose that for all h ∈ H, the map x → h |ψ x is a measurable function on X. Let A be a closed densely defined operator on H. Then the following are equivalent.
i) ψ is a continuous atomic system for A; ii) ψ is a continuous A-frame; iii) ψ is a Bessel function and there exists φ a Bessel function of H A such that Af |h = X f |φ x A ψ x |h dµ(x), ∀f ∈ D(A), ∀h ∈ H; iv) C * ψ is bounded and R(A) ⊂ R(C * ψ ); v) C * ψ is bounded and there exists M ∈ B(H A , L 2 (X, µ)) such that A = C * ψ M . vi) S ψ = C * ψ C ψ ≥ αAA ♯ on H, for some α > 0 and ψ is a Bessel function for H; vii) A = S 1/2 ψ U , for some U ∈ B(H A , H).
Note also that if A ∈ B(H), then the graph norm of A is defined on H and it is equivalent to · , thus our notion of continuous A-frame reduces to that of literature (see e.g. [2]).