Bochner pg-frames

In this paper we introduce the concept of Bochner pg-frames for Banach spaces. We characterize the Bochner pg-frames and specify the optimal bounds of a Bochner pg-frame. Then we define a Bochner qg-Riesz basis and verify the relations between Bochner pg-frames and Bochner qg-Riesz bases. Finally, we discuss the perturbation of Bochner pg-frames. MSC:42C15, 46G10.


Introduction and Preliminaries
The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer [9] in 1952 to study some deep problems in nonharmonic Fourier series.After the fundamental paper [7] by Daubechies, Grossmann and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames.Frames play a fundamental role in signal processing, image and data compression and sampling theory.They provided an alternative to orthonormal bases, and have the advantage of possessing a certain degree of redundancy.A discrete frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements.For more details about discrete frames see [6].Resent results show that frames can provide a universal language in which many fundamental problems in pure mathematics can be formulated: The Kadison-Singer problem in operator algebras, the Bourgain-Tzafriri conjecture in Banach space theory, paving Toeplitz operators in harmonic analysis and many others.Various types of frames have been proposed, for example, pg-frames in Banach spaces [2], fusion frames [4], continuous frames in Hilbert space [3], continuous frames in Hilbert spaces [16], continuous g-frames in Hilbert spaces [1], (p, Y )-operator frames for a Banach space [12].This paper is organized as follows.In Section 2, we introduce the concept of Bochner pg-frames for Banach spaces.Actually, continuous frames motivate us to introduce this kind of frames and analogous to continuous frames which are generalized version of discrete frames, we want to generalize pg-frames in a continuous sense.Like continuous frames, these frames can be used in those areas that we need generalized frames in a continuous aspect.Also, we define corresponding operators (synthesis, analysis and frame operators) and discuss their characteristics and properties.In Section 3, we define a Bochner qg-Riesz basis and verify its relations by Bochner pg-frames.Finally, Section 4 is devoted to perturbation of Bochner pg-frames.
Throughout this paper, X and H will be a Banach space and a Hilbert space, respectively, and {H ω } ω∈Ω is a family of Hilbert spaces.
Suppose that (Ω, Σ, µ) is a measure space, where µ is a positive measure.
The following definition introduces Bochner measurable functions.
If µ is a measure on (Ω, Σ) then X has the Radon-Nikodym property with respect to µ if for every countably additive vector measure γ on (Ω, Σ) with values in X which has bounded variation and is absolutely continuous with respect to µ, there is a Bochner integrable function g : Ω −→ X such that for every set E ∈ Σ.
A Banach space X has the Radon-Nikodym property if X has the Radon-Nikodym property with respect to every finite measure.Spaces with Radon-Nikodym property include separable dual spaces and reflexive spaces, which include, in particular, Hilbert spaces.Remark 1.3.Suppose that (Ω, Σ, µ) is a measure space and X * has the Radon-Nikodym property.Let 1 ≤ p ≤ ∞.The Bochner space of L p (µ, X) is defined to be the Banach space of (equivalence classes of) X-valued Bochner measurable functions F from Ω to X for which the norms In [8], [5] and [10, p.51] it is proved that if 1 ≤ p < ∞ and q is such that 1 p + 1 q = 1, then L q (µ, X * ) is isometrically isomorphic to (L p (µ, X)) * if and only if X * has the Radon-Nikodym property.This isometric isomorphism is the mapping where the mapping ψ(g) is defined on L p (µ, X) by So for all f ∈ L p (µ, X) and g ∈ L q (µ, X * ) we have In the following, we use the notation < f, g > instead of < f, ψ(g) >, so for all f ∈ L p (µ, X) and g ∈ L q (µ, X * ) Particularly, if H is a Hilbert space then (L p (µ, H)) * is isometrically isomorphic to L q (µ, H).So, for all f ∈ L p (µ, H) and g ∈ L q (µ, H) in which < f (ω), g(ω) > dose not mean the inner product of elements f (ω), where ν : H −→ H * is the isometric isomorphism between H and H * , for more details refer to [14, p.54].
We will use the following lemma which is proved in [11].
Lemma 1.4.If U : X −→ Y is a bounded operator from a Banach space X into a Banach space Y then its adjoint U * : Y * −→ X * is surjective if and only if U has a bounded inverse on R U .
Note that for a collection {H β } β∈B of Hilbert spaces, we can suppose that there exists a Hilbert space K such that for all β ∈ B, H β ⊆ K, where K = ⊕ β∈B H β is the direct sum of {H β } β∈B , see 3.1.5 in [15, p.81].

Bochner pg-frames
Bochner spaces are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation: if the temperature g(t, x) is a scalar function of time and space, one can write (f (t))(x) := g(t, x) to make f a function of time, with f (t) being a function of space, possibly in some Bochner space.Now, we intend to use this space to define a new kind of frames which contain all of continuous and discrete frames, in other words we will generalize the g-frames to a continuous case that is constructed on concept of Bochner spaces.Of course, this new frame can be useful in function spaces and operator theory to gain some general results that are achieved by g-frames or discrete frames.

Bochner pg-frames and corresponding operators
We start with the definition of Bochner pg-frames.Then we will give some characterizations of these frames.
is Bochner measurable, (ii) there exist positive constants A and B such that (2.1) A and B are called the lower and upper Bochner pg-frame bound, respectively.We call that {Λ ω } ω∈Ω is a tight Bochner pg-frame if A and B can be chosen such that A = B and a Parseval Bochner pg-frame if A and B can be chosen such that Example 2.2.Let {f i } i∈I be a frame for Hilbert space H, Ω = I and µ be a counting measure on Ω. Set So, {Λ ω } ω∈Ω is a Bochner pg-frame for L p (Ω) with respect to {H ω } ω∈Ω .Now, we state the definition of some common corresponding operators for a Bochner pg-frame.Definition 2.4.Let {Λ ω } ω∈Ω be a Bochner pg-Bessel family for X with respect to {H ω } ω∈Ω and q be the conjugate exponent of p.We define the operators T and U , by The operators T and U are called the synthesis and analysis operators of {Λ ω } ω∈Ω , respectively.
The following proposition shows these operators are bounded.Proof.Suppose that {Λ ω } ω∈Ω is a Bochner pg-Bessel family with bound B and q is the conjugate exponent of p.We show that for all x ∈ X and all For each n, < λ n , g n > is a simple function and Thus T is well-defined and T ≤ B. By a similar discussion, U is well-defined and U ≤ B.
The following proposition provides us with a concrete formula for the analysis operator.
Proof.Let q be the conjugate exponent of p and x ∈ X.For all G ∈ L q (µ, ⊕ ω∈Ω H ω ), we have The following proposition shows that it is enough to check the Bochner pg-frame conditions on a dense subset.The discrete version of this proposition is available in [6, Lemma 5.1.7].
Proposition 2.7.Suppose that (Ω, Σ, µ) is a measure space where µ is σfinite.Let {Λ ω ∈ B(X, H ω ) : ω ∈ Ω} be a family such that for each x ∈ X, ω −→ Λ ω (x) is Bochner measurable and assume that there exist positive constants A and B such that (2.1) holds for all x in a dense subset V of X.Then {Λ ω } ω∈Ω is a Bochner pg-frame for X with respect to {H ω } ω∈Ω with bounds A and B.
is a family of disjoint and measurable subsets of Ω.If {Λ ω } ω∈Ω is not a Bochner pg-Bessel family for X then there exists x ∈ X such that and there exist finite sets I and J such that The assumption implies that which is a contradiction to (2.4)(by the Lebesgue's Dominated Convergence Theorem).So {Λ ω } ω∈Ω is a pg-Bessel family for X with respect to {H ω } ω∈Ω and Bessel bound B. Now, we show that Since {Λ ω } ω∈Ω is a pg-Bessel family for X, the operator U defined by (2.3) is well defined and bounded.Assume that q is the conjugate exponent of p and let It is obvious that for each k ∈ N, G k and G belong to L q (µ, ⊕ ω∈Ω H ω ), and we have so by the Lebesgue's Dominated Convergence Theorem By letting x k → x, the proof is completed.

Characterization of Bochner pg-frames
Now we give some characterizations of Bochner pg-frames in terms of their corresponding operators.
At first, we show the next lemma that is very useful in case of complex valued L p -spaces.
Lemma 2.8.Let (Ω, Σ, µ) be a measure space where µ is σ-finite.Let 1 < p < ∞ and q be its conjugate exponent.If F : Ω −→ H is Bochner measurable and for each G ∈ L q (µ, H), n=1,m=1 is a family of disjoint and measurable subsets of Ω.We have where Then G is Bochner measurable, and F (ω) p dµ(ω)) which is a contradiction.
Proof.Let q be the conjugate exponent of p and for x ∈ X, consider Similar to discrete frames, the analysis operator has closed range.
Proof.(i) Since {Λ ω } ω∈Ω is a Bochner pg-frame, by Proposition 2.5, T is well-defined and bounded.From the proof of Lemma 2.10, U is bounded below.So, by Lemma 1.4 and Lemma 2.12(i), U * = T is surjective.(ii) Since T is bounded, {Λ ω } ω∈Ω is a Bochner pg-Bessel family, by Theorem 2.9.Since T = U * is surjective, U has a bounded inverse on R U by Lemma 1.4.So there exists A > 0 such that for all x ∈ X, U x p ≥ A x .By Proposition 2.6, for all x ∈ X A Hence {Λ ω } ω∈Ω is a Bochner pg-frame.
Corollary 2.14.If {Λ ω } ω∈Ω is a Bochner pg-frame for X with respect to {H ω } ω∈Ω and q is the conjugate exponent of p then for each x * ∈ X * there exists G ∈ L q (µ, ⊕ ω∈Ω H ω ) such that Proof.It is obvious.
The optimal Bochner pg-frame bounds can be expressed in terms of synthesis and analysis operators.
Theorem 2.15.Let {Λ ω } ω∈Ω be a Bochner pg-frame for X with respect to {H ω } ω∈Ω .Then T and Ũ are the optimal upper and lower Bochner pgframe bounds of {Λ ω } ω∈Ω , respectively, where Ũ is the inverse of U on R U and T , U are the synthesis and analysis operators of {Λ ω } ω∈Ω , respectively.
Proof.From the proof of Theorem 2.9, for each x ∈ X, we have Next theorem presents some equivalent conditions for a Bochner pgframe being a Bochner qg-Riesz basis.Theorem 3.3.Suppose that (Ω, Σ, µ) is a measure space where µ is σ-finite.Let {Λ ω } ω∈Ω be a Bochner pg-frame for X with respect to {H ω } ω∈Ω with synthesis operator T and analysis operator U and q be the conjugate exponent of p. Then the following statements are equivalent: Proof.(i) → (ii): It is obvious.(ii) → (i): By Theorem 2.13(i), the operator T defined by (2.2) is bounded and onto.By (ii), T is also injective.Therefore T has a bounded inverse T −1 : X * −→ L q (µ, ⊕ ω∈Ω H ω ) and hence {Λ ω } ω∈Ω is a Bochner qg-Riesz basis for X * .(i) → (iii): By Theorem 3.2, T is invertible, so T * is invertible.Lemma 2.12(ii) implies that R U = L p (µ, ⊕ ω∈Ω H ω ).(iii) → (i): Since the operator U is invertible, by Lemma 2.12 , T = U * is invertible.
From (4.1) and (4.3), we obtain that For a given x ∈ X, there exists x * ∈ X * such that x * = 1, x = x * (x). Hence