Frames for Operators in Banach Spaces

Abstract

A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces.

1 Introduction

Frames are a tool for the construction of series expansions in Hilbert spaces. Frames provide stable expansions, quite in contrast to orthogonal expansions — they may be overcomplete and the coefficients in the frame expansion therefore need not be unique. The redundancy and flexibility offered by frames has spurred their applications in a variety of areas throughout mathematics and engineering, such as operator theory [11], harmonic analysis [9], pseudo-differential operators [10], quantum computing [5], signal and image processing [4], and wireless communication [13].

Theoretical research of frames for Banach spaces is quite different from that for Hilbert spaces. Due to the lack of an inner product, the properties of Hilbert frames usually do not transfer automatically to Banach spaces. Gröchenig [8] generalized Banach frames with respect to certain sequence spaces. The main feature of frames that Gröchenig was trying to capture in a general Banach space was the unique association of a vector in a Hilbert space with the natural set of frame coefficients. After the work of Gröchenig, frames in Banach spaces have become topic of investigation for many mathematicians.

A sequence space X _d is called a BK-space if it is a Banach space and the coordinate functionals are continuous on X _d . If the canonical unit vectors form a Schauder basis for X _d , then X _d is called a CB-space and its canonical basis is denoted by {e _n }. If X _d is reflexive and a CB-space, then X _d is called an RCB-space. Also, the dual of X _d is denoted by ${ X}_{d}^{*}$. When ${ X}_{d}^{*}$ is a CB-space, then its canonical basis is denoted by ${\{e_{n}^{*}}\}$.

We denote by $\mathcal B(X)$ the space of all bounded linear operators on a Banach space X. For $T \in \mathcal B(X)$, we denote D(T), R(T), and N(T) for domain, range and nullspace of T, respectively. The set of all natural numbers is denoted by N. For simplicity, a sequence {f _n : n ∈ N} indexed by N will be abbreviated as {f _n } throughout the paper.

The results in this paper are organized as follows. In Section 2, we recall basic definitions, known results on K-frames in Hilbert spaces and X _d -frames in Banach spaces. Two new notions, “atomic systems” and “frames for operators” are defined in Section 3, without using semi-inner products. Operators preserving them and generating new such frames using old ones have been discussed. In the end, it is shown that the frames for operators in Banach spaces share few nice properties of frames for operators in Hilbert spaces, under some crucial assumptions. Throughout the paper, all spaces are nontrivial; operators are non-zero, and X is a reflexive separable Banach space.

2 Notations and Preliminaries

Găvruţa [7] introduced two notions, “atomic systems” and “K-frames” in a separable Hilbert space H, as a generalization of families of local atoms [6], where $K\in \mathcal B(H)$.

Definition 1

[7] A sequence {f _n } in H is called an atomic system for K, if the following conditions are satisfied:

• 1. the series $\displaystyle \sum\limits_{n} c_{n} f_{n}$ converges for all c = {c _n }∈ ℓ ₂ ;

• 2. there exists C > 0 such that for every f ∈ H there exists a _f = {a _n }∈ ℓ ₂ such that $\| a_{f} \|_{\ell _{2}} \leq C \| f \|$ and ${Kf}={\sum }_{n} a_{n} f_{n}$.

The condition 1 in Definition 1 says that {f _n } is a Bessel sequence.

Definition 2

[7] Let $K\in \mathcal B(H)$. A sequence {f _n } in H is called a K-frame for H if there exist two constants 0 < λ ≤ μ < ∞ such that

$$\lambda {\| K^{*}f \|}^{2} \leq \sum\limits_{n}|\langle f,f_{n} \rangle|^{2} \leq \mu {\| f \|}^{2} \text{ for all }f \in H. $$

The constants λ and μ are called the lower and upper bounds respectively, for the K-frame {f _n }. If the above inequalities hold only for $f\in \overline {span} {\{f_{n}}\}$, then {f _n } is said to be a K-frame sequence.

If K is equal to I, the identity operator on H, then K-frames and K-frame sequences are just ordinary frames and frame sequences, respectively. It is proved that these two concepts are equivalent [7]. Because of the higher generality of K-frames, many properties for ordinary frames may not hold for K-frames, such as the corresponding synthesis operator for K-frames is not surjective, the frame operator for K-frames is not isomorphic, the alternate dual reconstruction pair for K-frames is not interchangeable in general. Also, the frame operator S for a K-frame is semidefinite, so there is also S ^1/2, but not positive. In general, it is not invertible. For more details on K-frames, see [7, 12, 14, 15, 17].

The concept of a family of local atoms in a Banach space X with respect to a BK-space X _d was introduced by Dastourian and Janfada [3] using a semi-inner product. This concept was generalized to an atomic system for an operator $K\in \mathcal B(X)$ called ${ X}_{d}^{*}$ -atomic system and it has been led to the definition of a new frame with respect to the operator K, called ${ X}_{d}^{*}$-K -frame. Unlike the traditional way of considering sequences in the dual space X ^∗, sequences in the original space X are considered in [3] to study a family of $X_{d}^{*}$-local atoms and $X_{d}^{*}$-atomic systems by making use of semi-inner products.

Appropriate changes have been made in the definitions of $X_{d}^{*}$-atomic systems and $X_{d}^{*}$-K-frames to fit them for sequences in the dual space without using semi-inner products, called X _d -atomic systems and X _d -K -frames, respectively. Thus, the notion of X _d -K-frames for Banach spaces can be thought of a generalization of X _d -frames. We start with the definition of an X _d -frame defined by Casazza, Christensen, and Stoeva [2] which is a natural generalization of Hilbert frames to Banach frames.

Definition 3

Let X be a Banach space and let X _d be a BK-space. A sequence {g _n } of elements in X ^∗, which satisfies

• 1. {g _n (f)}∈ X _d for all f ∈ X,

• 2. There are constants 0 < λ ≤ μ < ∞ such that for each f ∈ X

  $$ \lambda \| f \|_{X} \leq \|\{ g_{n} (f) \} \|_{X_{d}} \leq \mu \| f \|_{X} $$

  (1)

is called an X _d -frame for X. The constants λ and μ are called lower and upper bounds respectively for {g _n }. When {g _n } satisfies the condition 1 and the upper inequality in (1) for all f ∈ X, {g _n } is called an X _d -Bessel sequence for X.

Note that the definition of X _d -frame is a part of the definition of a Banach frame introduced by Gröchenig [8]. If X is a Hilbert space and X _d = ℓ ₂, the X _d -frame inequalities in (1) mean that {g _n } is a frame, and in this case it is well-known that there exists a sequence {f _n } in X such that for each f ∈ X,

$$f=\sum\limits_{n} \langle f, f_{n}\rangle \ g_{n} =\sum\limits_{n} \langle f, g_{n}\rangle \ f_{n}.$$

Similar reconstruction formulas are not always available in the Banach space setting.

Lemma 1

[2] Let X _d be a BK-space for which the canonical unit vectors {e _n } form a Schauder basis. Then the space $Y_{d}= \{F(e_{n}) : F\in X_{d}^{*} \}$ with the norm $\|\{F(e_{n})\}\|_{Y_{d}}=\|F\|_{X_{d}^{*}}$ is a BK-space isometrically isomorphic to $X_{d}^{*}$. Also, every continuous linear functional F on X _d has the form $F(c)={\sum }_{n} c_{n}d_{n}$, where {d _n } = F(e _n ), is uniquely determined by d _n = F(e _n ), and $\|F\|_{X_{d}^{*}}=\|\{d_{n}\}\|_{Y_{d}}$.

Lemma 2

[2] Let X _d be a BK-space and let ${ X}_{d}^{*}$ be a CB-space. If {g _n } ⊆ X ^∗ is an X _d -Bessel sequence for X with bound μ, then the operator $L: \{ d_{n} \} \mapsto {\sum }_{n} d_{n} g_{n}$ is well-defined (hence bounded) from ${ X}_{d}^{*}$ into X ^∗ and ∥L∥≤ μ. If X _d is reflexive, the converse is also true.

Let X _d be a BK-space and let {g _n } be a sequence in X ^∗. If {g _n } satisfies only the upper inequality in (1), the analysis operator U from X to X _d mapped by f↦{g _n (f)}, is well-defined and linear, having domain D(U) = {f ∈ X : {g _n (f)} ∈ X _d }. The domain D(U) is a subspace (not necessarily closed) of X. If {g _n } is an X _d -Bessel sequence for X, then D(U) = X and U is bounded with the norm ∥U∥≤ μ.

If only the lower inequality in (1) is satisfied by {g _n }, then U is bounded below on D(U). Thus if {g _n } satisfies the X _d -frame inequalities in (1), we get that U is bounded and bounded below on D(U). Hence R(U) is closed in X _d and the inverse U ⁻¹ : R(U) → D(U) is also bounded with the norm $\| U^{-1} \| \leq \frac {1}{\lambda }$. We can conclude that given an X _d -frame {g _n }⊆ X ^∗ for X, the analysis operator U : X → X _d defined by U f = {g _n (f)} is an isomorphism of X onto R(U).

Given a sequence {g _n } in X ^∗, we now consider a function $L:X_{d}^{*} \to X^{*}$, called the synthesis operator, mapped as $\{d_{n}\}\mapsto {\sum }_{n} d_{n} g_{n}$ is well-defined and linear on the domain $D(L)=\left \{ \{d_{n}\} \in X_{d}^{*} : {\sum }_{n} d_{n} g_{n} \text { converges in } X^{*} \right \}.$ If {g _n }⊆ X ^∗ is an X _d -Bessel sequence in X with bound λ and if $X_{d}^{*}$ is a CB-space, then L is bounded from $X_{d}^{*}$ to X ^∗ and ∥L∥≤ μ, by Lemma 2. If X _d is a CB-space, then U ^∗ = L. If X _d is reflexive and {g _n } is an X _d -frame for X, then U = L ^∗ because X is isomorphic to a closed subspace of X _d and every closed subspace of a reflexive space is reflexive. Hence X is also reflexive. The section ends with a result connecting majorization, factorization and range inclusion for operators on Banach spaces.

Theorem 1

[1] Let X, Y, Z be Banach spaces and let $A\in \mathcal B(X, Y), B\in \mathcal B(Z, Y)$. Then the following statements hold:

• 1. If A = B T for some $T\in \mathcal B(X, Z)$, then B ^∗ majorizes A ^∗. The converse is true when N(B) is complemented in Z, and Z is reflexive. Note that B ^∗ majorizes A ^∗ if there exists C > 0 such that for each f ∈ Y, ∥A ^∗ f∥ _X ≤ C∥B ^∗ f∥ _Z .

• 2. If R(A) ⊆ R(B), then B ^∗ majorizes A ^∗. The converse is true when Z is reflexive.

3 Atomic System for Banach Spaces

Definition 4

Let X be a Banach space and let X _d be a BK-space. Let $K\in {\mathcal B(X^{*})}$ and {g _n }⊆ X ^∗. We say that {g _n } is an X _d -atomic system for X with respect to K if the following statements hold:

• 1. $\displaystyle \sum\limits_{n} d_{n} g_{n}$ converges in X ^∗ for all d = {d _n } in ${ X_{d}^{*}}$ and there exists μ > 0 such that $\Big \|\displaystyle \sum\limits_{n} d_{n} g_{n}\Big \|_{X^{*}} \leq \mu \|d\|_{X_{d}^{*}}$;

• 2. there exists C > 0 such that for every g ∈ X ^∗ there exists a _g = {a _n }∈ X _d such that $\| a_{g} \|_{{ X}_{d}} \leq C \| g \|_{X^{*}}$ and $Kg = {\sum }_{n} a_{n} g_{n}$.

When X _d is reflexive, the condition 1 in Definition 4 actually says that {g _n } is an X _d -Bessel sequence for X with bound μ, by Lemma 2. We find a necessary condition for a sequence {g _n }⊆ X ^∗ to be an X _d -atomic system for X with respect to a given operator K if the associated sequence space satisfies the following crucial property : For each {f _n },{h _n } ∈ X _d ,

$$ \Big| \sum\limits_{n} f_{n} h_{n} \Big| \leq \|\{f_{n}\}\|_{X_{d}} \|\{h_{n}\}\|_{X_{d}}. $$

(2)

For instance, let {f _n },{h _n } ∈ ℓ _p and p ∈ (1,2]. Then the conjugate of p, q lies in [2, ∞). Hence by Hölder’s inequality, the sequence space ℓ _p for 1 < p ≤ 2 satisfies the inequality (2).

Theorem 2

Let X _d be a BK-space. Let {g _n } be a sequence in X ^∗ and $K\in \mathcal B(X^{*})$. If {g _n } is an X _d -atomic system for X with respect to K and the sequence space X _d satisfies the inequality (2), then there exists a constant λ > 0 such that

$$\| K^{*} f \|_ X \leq \lambda \| {\{ g_{n} (f)}\}\|_{{ X}_{d}} \ \text{ for each } f\in X.$$

Proof

Suppose {g _n } is an X _d -atomic system for X with respect to K. Then there is some C > 0 such that for every g ∈ X ^∗ there exists a _g = {a _n } ∈ X _d such that $\| a_{g} \|_{{ X}_{d}} \leq C \ \| g \|_{X^{*}}$ and $Kg ={\sum }_{n} a_{n} g_{n}$. Hence for each f ∈ X,

$$\begin{array}{@{}rcl@{}} \| K^{*} f \|_{X} & =& \sup_{g\in X^{*}, \ \| g \| = 1} | g(K^{*} f) | \\ & =& \sup_{g\in X^{*}, \ \| g \| = 1} \ | (Kg) (f) | \\ & = &\sup_{g\in X^{*}, \ \| g \| = 1} \Big| \sum\limits_{n} a_{n} g_{n} (f) \Big| \\ & \leq & \sup_{g\in X^{*}, \ \| g \| = 1} \ \| {\{a_{n}}\} \|_{{ X}_{d}} \ \| \{ g_{n} (f) \} \|_{{{ X}_{d}}} \\ & =& \sup_{g\in X^{*}, \ \| g \| = 1} \ \| a_{g} \|_{{{ X}_{d}}} \ \| \{ g_{n} (f)\} \|_{{{ X}_{d}}} \\ & \leq & C \sup_{g\in X^{*}, \ \| g \| = 1} \ \|g\|_{X^{*}} \ \| \{ g_{n} (f) \} \|_{{ X}_{d}} \quad \text{[using \ } \| a_{g} \|_{{ X}_{d}} \leq C \ \| g \|_{X^{*}}]. \end{array} $$

Thus for some C > 0, $\| K^{*} f \|_{X} \leq C \ \| \{ g_{n} (f) \} \|_{{ X}_{d}}$ for each f ∈ X. □

Definition 5

Let X be a Banach space and let X _d be a BK-space. Let $K\in \mathcal B(X^{*})$ and {g _n }⊆ X ^∗. We say that {g _n } is an X _d - K-frame for X if the following statements hold:

• 1. {g _n (f)}∈ X _d for each f ∈ X;

• 2. there exist two constants 0 < λ ≤ μ < ∞ such that

  $$\lambda \| K^{*} f \|_ X \leq \| {\{ g_{n} (f)}\} \|_{{ X}_{d}} \leq \mu \| f \|_ X \text{ for each } f\in X .$$

The elements λ and μ are called the lower and upper X _d -K-frame bounds.

We say that an X _d -frame for X is an X _d -I-frame for X, where I is the identity operator on X ^∗. The set of all X _d -frames for X can be considered as a subset of X _d -K-frames for X. Thus X _d -K-frame is a generalization of X _d -frame for a Banach space X. We present an example for an X _d -K-frame which is not an X _d -frame for X.

Example 1

Let X be the space of all triplets (α ₁, α ₂, α ₃) with complex scalars and having 3/2-norm, denoted by ℓ _3/2(3). Let {g _n }⊆ X ^∗ = ℓ ₃(3) be such that for n = 1,2,3, g _n (e _m ) = δ _{n m} , where e _m ’s are vectors in X, having 1 in m ^th place and 0 elsewhere, and g _n = 0 for all n ≥ 4. Define K : X ^∗→ X ^∗ as follows:

$$Kg_{1}=0, \ Kg_{2}=g_{3}, \text{ and } Kg_{3}=g_{2}.$$

For any f ∈ X, we have $f={\sum }_{n=1}^{3} \alpha _{n}e_{n}$ and

$$\|K^{*}f\|_{X}=\|\alpha_{2}e_{3}+\alpha_{3}e_{2}\|_{3/2}=\big(|\alpha_{2}|^{3/2} +|\alpha_{3}|^{3/2}\big)^{2/3}= \| {\{ g_{n} (f)}\}_{n=2}^{\infty} \|_{\ell_{3/2}}.$$

Then $\{g_{n}\}_{n=2}^{\infty }$ is an X _d -K frame for X. But it is not an X _d -frame because there is no constant λ such that for any scalar α ₁,

$$\lambda \|f\|_{X}=\big(|\alpha_{1}|^{3/2}+|\alpha_{2}|^{3/2} +|\alpha_{3}|^{3/2}\big)^{2/3} \leq \big(|\alpha_{2}|^{3/2} +|\alpha_{3}|^{3/2}\big)^{2/3}=\| {\{ g_{n} (f)}\}_{n=2}^{\infty} \|_{\ell_{3/2}}.$$

We can generate new X _d -K-frames for X from each X _d -frame for X and each operator $K\in \mathcal B(X^{*})$, by the following proposition.

Proposition 1

If {g _n } is an X _d -frame for X and $K\in \mathcal B(X^{*})$, then {K g _n } is an X _d -K-frame for X.

Proof

Suppose {g _n } is an X _d -frame for X. Then {g _n (f)}∈ X _d , for all f ∈ X and there are constants 0 < λ ≤ μ < ∞ such that for each f ∈ X

$$\lambda \| f \|_{X} \leq \|\{ g_{n} (f) \} \|_{X_{d}} \leq \mu \| f \|_{X}. $$

Let f ∈ X be fixed. Since (K g _n )(f) = g _n (K ^∗ f) and K ^∗ f ∈ X, we have {(K g _n )(f)}∈ X _d . Also, ∥K ^∗ f∥ _X ≤∥K∥ ∥f∥ _X gives that for each f ∈ X,

$$\lambda \|K^{*}f\|_{X} \leq \|\{(Kg_{n}) (f) \} \|_{X_{d}} \leq \mu \|K\| \ \| f \|_{X}.$$

Thus {K g _n } is an X _d -K-frame for X. □

The following example illustrates that an X _d -Bessel sequence is an X _d -K-frame but it is not the same for the other operator T.

Example 2

Let X = ℓ _3/2(3). Let {g _n }⊆ X ^∗ = ℓ ₃(3) be such that for n = 1,2,3, g _n (e _m ) = δ _{n m} , and g _n = 0 for all n ≥ 4. Define K and T from X ^∗ to X ^∗ as follows: K g ₁ = 0, K g ₂ = g ₃, and K g ₃ = g ₂, and T g ₁ = g ₁, T g ₂ = g ₃, and T g ₃ = g ₂. Then $\{g_{n}\}_{n=2}^{\infty }$ is an X _d -K frame but it is not an X _d -T-frame for X.

Theorem 3

Let {g _n } be an X _d -K-frame for X. Let $T\in \mathcal B(X^{*})$ be such that R(T) ⊆ R(K). Then {g _n } is an X _d -T-frame for X.

Proof

Suppose {g _n } is an X _d -K-frame for X. Then there are constants 0 < λ ≤ μ < ∞ such that for each f ∈ X

$$ \lambda \|K^{*} f \|_{X} \leq \|\{ g_{n} (f) \} \|_{X_{d}} \leq \mu \| f \|_{X}. $$

(3)

Since R(T) ⊆ R(K), by Theorem 1, there exists C > 0 such that for each f ∈ X, ∥T ^∗ f∥ _X ≤ C∥K ^∗ f∥ _X . From the second inequality in (3), we have for each f ∈ X

$$\frac{\lambda}{C} \| T^{*}f \|_{X} \leq \|\{ g_{n} (f) \} \|_{X_{d}} \leq \mu \| f \|_{X}.$$

Hence {g _n } is an X _d -T-frame for X. □

Theorem 4

Let X _d be a reflexive space and let {g _n } ⊆ X ^∗. Let {e _n } be the canonical unit vectors for X _d and $X_{d}^{*}$. Then {g _n } is an X _d -K-frame for X if and only if there exists a bounded linear operator $L: X^{*}_{d} \to X^{*}$ such that L e _n = g _n and R(K) ⊆ R(L).

Proof

Since {g _n } is an X _d -K-frame for X, there exist constants 0 < λ ≤ μ < ∞ such that for each f ∈ X,

$$\lambda \| K^{*} f \|_ X \leq \| {\{g_{n} (f)}\} \|_{{ X}_{d}} \leq \mu \| f \|_ X .$$

Hence the operator U : X → X _d defined by U f = {g _n (f)} is bounded and ∥U∥≤ μ. The adjoint of U, $U^{*} : {{ X}_{d}^{*}} \to X^{*}$ satisfies (U ^∗ e _n )(f) = e _n (U f) = g _n (f). Since X _d is an RCB-space, U ^∗ = L, hence we get L e _n = g _n . Also we have

$$\lambda \| K^{*} f \|_{X} \leq \| {\{g_{n} (f)}\} \|_{X_{d}}= \| L^{*} f \|_{X_{d}} \text{ for each } f\in X .$$

Thus by Theorem 1, R(K) ⊆ R(L).

On the other hand, suppose there exists a bounded linear operator $L: X^{*}_{d} \to X^{*}$ such that L e _n = g _n and R(K) ⊆ R(L). Then by Theorem 1, there exists λ > 0 such that for each f ∈ X, $\lambda \|K^{*} f\|_{X} \leq \|L^{*}f\|_{X_{d}}$. Thus for each f ∈ X,

$$\lambda \|K^{*} f\|_{X} \leq \|\{g_{n}(f)\}\|_{X_{d}}=\|L^{*}f\|_{X_{d}}\leq \|L\| \ \|f\|_{X}. $$

(4)

□

Corollary 1

Let X _d be a reflexive space and let {g _n } ⊆ X ^∗. Let {e _n } be the canonical unit vectors for X _d and $X_{d}^{*}$. Let N(L) be complemented in $X_{d}^{*}$. Then {g _n } is an X _d -K-frame for X if and only if L = K V for some $V\in \mathcal B(X_{d}^{*}, K^{*}).$

Zhang and Zhang [16] defined frames in Banach spaces via a compatiable semi-inner product, which is a natural substitute for inner products on Hilbert spaces. As assumed in the paper [16], we assume that X _d is reflexive, the canonical unit vectors {e _n } form a Schauder basis for X _d and $X_{d}^{*}$; the following crucial requirement is also imposed as in [16]: If d = {d _n } is a sequence of scalars satisfying ${\sum }_{n} c_{n} d_{n}$ converges for every c = {c _n } ∈ X _d , then $d\in X_{d}^{*}$, and if the above series converges for all $d\in X_{d}^{*}$, then c ∈ X _d .

For instance, if X _p = ℓ _p ,1 < p < ∞, then $X_{d}^{*}=\ell _{q}$, where $\displaystyle \frac {1}{p}+\frac {1}{q}=1$ , it satisfies all of our requirements on X _d and $X_{d}^{*}$. The above requirements about the spaces X and X _d are assumed in the rest of the paper. We now prove the converse of Theorem 2 with the above assumptions.

Theorem 5

Let X be a Banach space and let X _d be a BK-space. Let {g _n } ⊆ X ^∗ be an X _d -Bessel sequence for X, and $K\in \mathcal B(X^{*})$. If N(L) is complemented, and if there exists a constant λ > 0 such that for each f ∈ X

$$\| K^{*} f \|_ X \leq \lambda \| {\{g_{n} (f)}\} \|_{{ X}_{d}},$$

then {g _n } is an X _d -atomic system for X with respect to K.

Proof

Using the synthesis operator L, the given inequality in hypothesis can be written as

$$\|K^{*}f\|_{X} \leq \lambda \|L^{*}f\|_{X_{d}} \ \text{ for all } f\in X.$$

By Theorem 1, K = L T for some $T\in \mathcal B(X^{*}, X_{d}^{*}).$ Let g ∈ X ^∗ be fixed. Then $Tg\in X_{d}^{*}$. Since X _d has the canonical unit vectors {e _n } as a Schauder basis, the continuous linear functional T g on X _d has the form $Tg(c) ={\sum }_{n} c_{n}d_{n}$, where {d _n } ∈ X _d is uniquely determined d _n = F(e _n ), and $\|Tg\|_{X_{d}^{*}}=\|\{d_{n}\}\|_{X_{d}}$. Since T is bounded, the sequence {d _n } associated for g ∈ X ^∗ satisfies

$$\|\{d_{n}\}\|_{X_{d}} = \|Tg\|_{X_{d}^{*}}\leq \|T\| \ \|g\|_{X^{*}}.$$

Also, we have

$$Kg=LTg=L(\{d_{n}\}) = \sum\limits_{n} d_{n}g_{n}.$$

Thus {g _n } is an X _d -atomic system for X with respect to K. □

Theorem 6

Let $K_{1} , K_{2} \in \mathcal B(X^{*})$. Let {g _n } be an X _d -atomic system for X with respect to K ₁, K ₂, and let α, β be scalars. If N(L) is complemented, then {g _n } is an X _d -atomic system for α K ₁ + β K ₂.

Proof

Suppose {g _n } is an X _d -atomic system for X with respect to K ₁, K ₂ and α, β are any scalars. Then there are constants 0 < λ _i ≤ μ _i < ∞ (i = 1,2) such that for each f ∈ X

$$\lambda_{i} \| K_{i}^{*} f \|_ X \leq \| {\{ g_{n} (f)}\} \|_{{ X}_{d}} \leq \mu_{i} \| f \|_ X.$$

By simple calculations, we get

$$\Big(\frac{|\alpha|}{\lambda_{1}}+\frac{|\beta|}{\lambda_{2}}\Big)^{-1} \|(\alpha K_{1}+\beta K_{2})^{*}f\|_{X} \leq \|\{g_{n}(f)\}\|_{X_{d}}\leq \Big(\frac{\mu_{1} + \mu_{2}}{2} \Big)\|f\|_{X}.$$

Therefore by Theorem 5, {g _n } is an atomic system for α K ₁ + β K ₂.

□

We now prove that the notions “atomic systems” and “frames for operators” are equivalent under the crucial assumptions. The proof of the result given below follows from Theorem 2 and Theorem 5.

Theorem 7

Let X _d be a sequence space satisfying the inequality (2) and let {g _n } ⊆ X ^∗ be an X _d -Bessel sequence for X. Let N(L) be complemented and $K\in \mathcal B(X^{*})$. Then the following statements are equivalent:

• 1. {g _n } is an X _d -atomic system for X with respect to K.

• 2. {g _n } is an X _d -K-frame for X.

Corollary 2

[7] Let {f _n } be a sequence in a Hilbert space H and let $K\in \mathcal B(H).$ Then the following statements are equivalent:

• 1. {f _n } is an atomic system for K.

• 2. {f _n } is a K-frame for H.

Proof

The proof follows from Theorem 7 because the assumptions are “redundant” if X is considered to be a Hilbert space with the sequence space X _d = ℓ ₂ in Theorem 7. □