On a Generalization of the Hilbert Frame Generated by the Bilinear Mapping

Abstract

The concept of -frame which is a generalization of the frame in Hilbert spaces generated by the bilinear mapping is considered. -frame operator is defined; analogues of some well-known results of frame theory are obtained in Hilbert spaces. Conditions for the existence of -frame in Hilbert spaces are obtained; the relationship between the definite bounded surjective operator and -frame is also studied. The concept of -orthonormal -basis is introduced.

1. Introduction

Frames for a Hilbert space were formally defined by Duffin and Schaeffer to study some problems in nonharmonic Fourier series [1]. Frames are applied in various branches of natural sciences, such as in signal processing, in the image processing, and in data compression. More details about the application of frames can be found, for example, in [2–5]. Numerous works have been dedicated to frames (see, e.g., [6–8]). More details about frame theory can be found in the monographs [9, 10] and also in review articles [11, 12]. One of the important facts of the theory of frames is the decomposition of each vector with respect to the frame in Hilbert space. It is a remarkable fact that in this case a sequence of coefficients has a minimal property. In frame theory, an important tool for obtaining frames is the stability of frames. Stability of frames in Hilbert spaces is studied in [13, 14] and so forth. Extension of the concept of frame to Banach spaces belongs to Gröchenig [15]. The concepts of a Banach frame and an atomic decomposition are introduced in [15]. Frames in Banach spaces have been also studied in [16–18].

The concept of -frame as a generalization of the frame in Hilbert spaces has been considered by Sun in [19]. Analogues of most properties of frames were obtained for the -frame in [20, 21]. Another generalization of the frame in Hilbert spaces is a concept of a -frame which is introduced in the works [22, 23] by the bilinear mapping. The concept of -frame is introduced and studied in [22], where is tensor mapping. Let us note that the approximate concepts associated with the linear mapping and related results have been introduced in [24, 25]. An atomic decomposition of Lebesgue spaces in the trigonometric systems with degenerate coefficients has been studied in [24, 25].

This paper is devoted to the study of the characteristic properties of -frame in Hilbert spaces. Analogues of the known results of the frame are established. Namely, a criterion of -frameness for the given sequence is established. The concept of -orthonormal -basis is introduced. In this paper, -frameness of the sequence, generated by the matrix operator, is studied in Hilbert spaces.

2. Some Notations and Auxiliary Facts

We will use the standard notation. Let and be Hilbert spaces with scalar products and , respectively. As usual, by we denote the space of all bounded linear operators acting from to . The kernel and the range of the operator are denoted by and , respectively. If and , then by Banach theorem . In the case when is closed, there exists such that , . Indeed, is a continuation of the operator by zero to all , where . The adjoint of the operator is denoted by .

By , denote the linear space of sequences of vectors from , with coordinatewise linear operations, such that . is a Hilbert space with the scalar productObviously, has a decomposition where is the Kronecker symbol.

Let be a Banach space with a norm . Consider the bilinear mapping , satisfying the conditionFor each pair and , let us consider the functional in , defined by the formula . Let us show that . Linearity of is obvious. Further, ; taking into account (3), we haveHence, . According to the Riesz theorem, there exists the unique element such that . Further, this element will be called the -scalar product of and and will be denoted by . It is clear that is linear and continuous for . It directly follows from that

We will also need some concepts and statements from [6, 22, 23].

Definition 1. Sequence is said to be -orthonormal in , if

By , denote the set of all finite sums of the form , .

Definition 2. Sequence is said to be -complete in , if the closure coincides with .

Let us provide the following criterion for -completeness of the sequence.

Theorem 3. Sequence is -complete in if and only if the equality , , implies .

Proof. Let the sequence be -complete in . Let us show that, from the equality , it follows that . Assume the contrary that is let there exists an element such that , , but . Then, and , we have Thus, is orthogonal to , and hence, by its continuity, will be orthogonal to . Since coincides with , then, in particular, , and therefore , which contradicts the assumption.
Conversely, let the sequence be such that, from the equation , it follows that . Let us prove its -completeness in . If does not coincide with , then there exists a nonzero element such that , . Then, , we have and, consequently, , . Then, , and this gives a contradiction. Therefore, is -complete in .
The theorem is proved.

Definition 4. The sequence is said to be -basis in , if uniquely represented in the form

In the case of -basis, is an orthonormal; then, it is called a -orthonormal -basis in .

It is clear that if the system forms a -orthonormal -basis for , then has a unique representation Indeed, taking into account the continuity of -scalar product with respect to , from (9) we obtain

The following theorem is an analogue of the criterion for an orthonormal basis.

Theorem 5. Let be a -orthonormal sequence. Then, the following conditions are equivalent:(1) is -orthonormal -basis in .(2) is -complete in .(3), the -Parseval identity holds.

Proof. Let condition hold. Then, it is clear that coincides with ; that is, condition is fulfilled. The equivalence of conditions and follows from the relation It remains to show that condition follows from condition . For every , we haveHence, it is easy to see that if the system is -complete in , then condition is true.
The theorem is proved.

In theory of frames, often use the following theorem, which describes some properties of the adjoint operator.

Theorem 6. Let . Then,(1) and ;(2) is closed if and only if is closed;(3) is surjective if and only if , .

3. Main Results

Let and be a Hilbert space with scalar products and , respectively, and let be a Banach space with the norm . Suppose that is bilinear mapping and satisfies condition (3). Consider a sequence of vector from .

Definition 7. Sequence is called a -frame in , if there exist constants such that

Constants and are called the bounds of -frame. When the right-hand side of (14) is fulfilled, then the sequence is called -Besselian in with a bound .

Let us provide some examples to -frames.

Example 8. Let , be Hilbert spaces and . Assume , where and . Let us recall that (see. [16]) the sequence is called -frame in , if there exist constants such that Then, -frame in is -frame in .

Example 9. Let , be Hilbert spaces and is their Hilbert tensor product. Assume that , where is a tensor product of elements and . Then, -frame in forms a -frame for [19].

Let us provide a characteristic property of -Besselian sequence in .

Theorem 10. Sequence is -Besselian in with a bound if and only if the bounded operator is defined:and . Moreover, an adjoint operator is determined by .

Proof. Let be -Besselian in with a bound . The series converges for every . Indeed, for , we haveSince the right-hand side tends to zero, as , then the sequence satisfies the Cauchy criterion and also converges. So the operator is defined. It is easy to show that Hence, we get .
Conversely, suppose that the bounded operator , is defined, and . Let us prove that the system is -Besselian in with a bound . For arbitrary and , consider . We haveHence, we directly obtain an expression for the adjoint operator We haveThe theorem is proved.

Let the sequence form a -frame for with the bounds and . Then, a bounded operator is defined by the following formula: Operator is called -frame operator for . Many of the properties of the ordinary frames are valid in this case.

Theorem 11. Let the sequence form a -frame for with bounds , and with -frame operator . Then is a positive, self-adjoint, bounded invertible operator and .

Proof. For every , we have that is, is a positive operator. Let the operator be defined by the expression . Then, by Theorem 10, it directly follows that , and therefore . Hence, we obtain that is a self-adjoint operator. Then, for every , the following inequality holds: and consequently . Thus, as surjective and injective operator, by Banach theorem, is a bounded invertible operator. Since holds, then from the relation it follows that .
Theorem is proved.

Theorem 12. Sequence forms a -frame for if and only if the bounded surjective operator, is defined.

Proof. Let the sequence form a -frame for and let be its -frame operator. Then, it is a -Besselian in and therefore, by Theorem 10, the bounded operator , , is defined. It remains to show that . Since and , then .
Conversely, suppose that a bounded surjective operator , is defined. Let us show that forms a -frame for . According to Theorem 10, the sequence is -Besselian in . Let us take an arbitrary . So ; then supposing , we haveThus,The theorem is proved.

Let us provide the criterion of -frameness with bounds and .

Theorem 13. Sequence forms a -frame for with the bounds and if and only if the following conditions are fulfilled:(1) is -complete in .(2)The bounded operator, is defined, such that

Proof. Let form a -frame for with bounds and . Let us prove the -completeness of the sequence in . Assume the contrary; that is, the sequence is not -complete in . Then, by Theorem 3, there exists a nonzero element such that , . According to (14), we obtain that , and hence the assumption is incorrect. Then, from the closedness of , closedness of follows. Therefore, . Let us take an arbitrary element . Let , such that . We haveHence, On the other hand,Conversely, suppose that conditions and hold. Let us show that is a surjective operator. Indeed, let and such that as . Then, there exists such that . From , it is easy to see that the sequence is fundamental and therefore converges to . Considering continuity of an operator , we obtain ; that is, is closed. Future, from -completeness of in and from , it follows that . As then, from , we have Consequently, . We have and thus we obtain andThe theorem is proved.

The following property of frames is a generalization of the similar properties with respect to the Riesz basis.

Theorem 14. Let the sequence form a -orthonormal -basis for . Then, the system forms a -frame for if and only if there exists a bounded surjective operator such that , and .

Proof. Let form a -frame for . By Theorem 12, the bounded surjective operator is defined: Consider the operator by the formula . It is clear that . Let us define the operator as follows: . It is obvious that is a bounded surjective operator and is fulfilled.
Conversely, suppose that there is a bounded surjective operator , such that , , . Let us show that is a -frame in . Let be an arbitrary element. We haveIndeed, we haveand therefore holds (40). Then,By virtue of the surjectivity of , there exists such that . On the other hand, . Thus, from (42) it follows that forms a -frame for .
Theorem is proved.

Let the sequence form a -frame for and consider the operator-matrix , , where . Assume that the system satisfies the following condition:

The following theorem is true, which of an independent interest.

Theorem 15. Let the sequence form a -frame for with bounds , and a -frame operator ; forms a -frame for with bounds and . If , , then is a bounded operator in .

Proof. Take .
We haveThus, the operator is bounded and .
The theorem is proved.

The following theorem in a certain sense is the inverse of Theorem 15.

Theorem 16. Let the sequence form a -frame for with bounds and , let be a bounded operator in , and the sequence is such that the relation (43) holds. Then, forms a -frame for if and only if

Proof. Let be a -frame in with the bounds and . Then, , we haveConversely, suppose that inequality (45) holds. Let us show that is -frame in . It is clear that Consequently,Then, according to (45) and (14), we obtain that is, forms a -frame for .
The theorem is proved.

The following theorem establishes a relationship between the frames and systems of systems and .

Theorem 17. Let the sequence form a -frame for with bounds and , the operators , , are such that the inequality is valid, , , and the following conditions are fulfilled: Then, is a -frame in with bounds and .

Proof. Take . Firstly, let us prove that is -Besselian in with a bound . We haveNow, let us establish for the left-hand side of inequality (14). We obtainConsider and , separately. We haveConsequently,The theorem is proved.

In particular, in the case of -frame, we obtain the following result.

Theorem 18. Let the sequence form a -frame for with bounds and , the operators , , are such that , , , and the following condition is fulfilled: Then, the bounded operators are defined in and are -frames in with bounds and .

Competing Interests

The authors declare that they have no competing interests.