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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 272407, 8 pages
Some New Characterizations and -Minimality and Stability of -Bases in the Hilbert Spaces
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
Received 15 March 2013; Accepted 27 May 2013
Academic Editor: Gestur Ólafsson
Copyright © 2013 Xunxiang Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The concept of -basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces. -basis plays the similar role in -frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties of -bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties of -bases which are related to -minimality. Finally, we obtain a perturbation result about -bases.
In 1946, Gabor  introduced a fundamental approach to signal decomposition in terms of elementary signals. In 1952, Duffin and Schaeffer  abstracted Gabor’s method to define frames in the Hilbert spaces. Frame was reintroduced by Daubechies et al.  in 1986. Today, frame theory is a central tool in many areas such as characterizing function spaces and signal analysis. We refer to [4–10] for an introduction to frame theory and its applications. The following are the standard definitions on frames in the Hilbert spaces. A sequence of elements of a Hilbert space is called a frame for if there are constants so that The numbers are called the lower (resp., upper) frame bounds. The frame is a tight frame if and a normalized tight frame if .
In , Sun raised the concept of -frame as follows, which generalizes the concept of frame extensively. A sequence is called a -frame for with respect to , which is a sequence of closed subspaces of a Hilbert space , if there exist two positive constants and such that for any We simply call a -frame for whenever the space sequence is clear. The tight -frame, normalized tight -frame, -Riesz basis are defined similarly. Recently, -frames in the Hilbert spaces have been studied intensively; for more details see [12–16] and the references therein.
In the history of the Hilbert space theory, the concept of basis was introduced more earlier than the concept of frame. And frames are generalizations of bases. When we look back the development of -frame theory, it is well known that -frames are generalizations of frames. Then it is naturally to ask what are the generalizations of bases in the similar manner? It is the motivation of our paper . In , we put up with the concept of -basis in the Hilbert space. We also find that -basis really fills in the gap in the -frame theory. -bases play the similar role in -frame theory to that the bases play in the frame theory. But we also find that the properties of basis cannot be adjusted trivially so that they are hold by -bases. For more details, please see . In this paper, we will do some study on -bases in the Hilbert spaces further. And some important properties of -bases are established. In Section 2, we will give the definitions and lemmas. In Section 3, we give some characterizations of -bases. In Section 4, we study the property of -minimal of -bases. In Section 5, we discuss the stability of -bases under perturbations.
Throughout this paper, we use to denote the set of all natural numbers, to denote the set of all integer numbers, and to denote the field of complex numbers. The sequence of always means a sequence of closed subspace of some Hilbert space . And all Hilbert spaces are complex separable Hilbert spaces.
2. Definitions and Lemmas
In this section, we introduce the definitions and lemmas which will be needed in this paper.
Definition 1. is called -complete with respect to if ,.
Definition 2. is called -linearly independent with respect to if ; then , where .
Definition 3. is called -minimal with respect to if, for any and any such that , we have .
Definition 4. Let denote the set of all maps from to . and are called biorthonormal if and are called -biorthonormal with respect to , if and are biorthonormal; that is,
Definition 5. We call a -basis for with respect to if for any there is a unique sequence with such that . The constant is called the basis constant associated with .
Remark 6. If is a Schauder basis of Hilbert space , then it induces a -basis of with respect to the complex number field , where is defined by . In fact, it is easy to see that for any , so, for any , there exists a unique sequence of constants such that .
Definition 7. If is a -basis of with respect to , then, for any , there exists a unique sequence such that and . We define a map , by for each . Then is well defined. We call it the -dual sequence of ; in case that is also a -basis, we call it the dual -basis of .
Lemma 8 (see ). is -complete if and only if .
The following result is about pseudoinverse, which plays an important role in some proofs.
Lemma 9 (see ). Suppose that is a bounded surjective operator. Then there exists a bounded operator (called the pseudoinverse of ) for which
The following is a simple property about -basis, which gives a necessary condition for -basis in terms of -complete and -linearly independent.
Lemma 10 (see ). If is a -basis for with respect to , then is -complete and -linearly independent with respect to .
3. Characterizations of -Bases
In this section, we characterize -bases under the condition that the sequence is bounded surjective operators.
Theorem 11. Suppose that and is surjective for each . Let is convergent}. If, for any , set , then is a Banach space, when is a -basis with respect to as well, , is a linear bounded and invertible operator; that is, is a homeomorphism between and .
Proof. Let ; then is convergent as . Hence is a convergent sequence, so it is bounded. So . It is obvious that, for ,, we have and . If , then, for any , . In particular, . Since is surjective, there exists a bounded operator such that by Lemma 9. So , which implies that . It implies that . So . Since , similarly, we have . Keep doing in this way; then we get for all . But is arbitrary, so . Thus is a norm on . Suppose is a Cauchy sequence, where . Then
For any fixed , we have that
Since is surjective, by Lemma 9. So . It implies that
So, for any fixed , is a Cauchy sequence. Suppose . From (8), we know that, for any , there exists , such that whenever , we have
Fix , since , so whenever ,
Since , is convergent. So there exists , such that whenever , we have . So when , we have
So is convergent; thus . Let in inequality (13), we get that . Hence is a complete normed space; that is, is a Banach space.
(2) If is a -basis, then it is -complete and -linearly independent with respect to by Lemma 10; then the operator is not only well defined but also is one to one and onto. And, for any , we have So is bounded operator. Since is a Banach space, by the open mapping theorem we get that is a homeomorphism.
Theorem 12. Suppose, for each , and is surjective. is a -basis of with respect to and is its -dual sequence. Then for all , let ; then ,, is a norm on and .
Proof. Let and be as defined in Theorem 11. Then for any , . So
Since , . Thus .
(3) It is obvious that is a seminorm. It is sufficient to show that . For any , we have On the other hand,
Theorem 13. Suppose that and is surjective. is a -basis for with respect to and is the -dual sequence of . Then is linear and bounded; that is, for each .
Proof. Suppose that and ; then and by Definition 7. Since , and . It implies that is linear, since, for any , we have Since is surjective, . It follows that So is bounded for each .
Theorem 14. Suppose that and is surjective. is a -basis for with respect to and is the dual sequence of . Then is also a -basis for and and are -biorthonormal.
Proof. By Theorem 13, we know that . Since is a -basis for with respect to and is the dual sequence of , , for all . In particular, for each , , we have Since the representation is unique, and for . It implies that So, and are -biorthonormal. Since , . So for every , we have that . To prove that is a -basis, it is sufficient to show that the representation is unique. It is sufficient to prove that is -linearly independent. Suppose that ; then, for each , we have It implies that is -linearly independent.
Theorem 15. Suppose that and is surjective. is a -basis for with respect to and is the dual -basis of . ; that is, is the partial sum operator associated with . Then under the norm defined in Theorem 12(3), , for all .
Theorem 16. Suppose that and is surjective. Then the following statements are equivalent: is a -basis for with respect to ; there exists such that is -biorthonormal with and is -complete and there exists such that is -biorthonormal with and is -complete and there exists such that is -biorthonormal with and
Proof. : It is obvious that the -dual basis is -biorthonormal with by Theorem 13 and, for each , .
: It is only needed, to show that the representation for is unique. It is sufficient to show that is -linearly independent. Suppose that , where , for all . Then , for each , since and are -biorthonormal. So is -linearly independent.
: It is obvious that is -complete. It is only needed to prove that, for all , . Since , . It implies that is convergent for each . Thus is a bounded sequence for each . Hence .
: It only needed, to show that . Since is a sequence of linear bounded operators on and, for each , . So by the Banach-Steinhaus theorem.
: For all , suppose that , . Since is linear and and are -biorthonormal, for , we have It implies that, for any , . Next we prove that, for all , it is also correct. Since is -complete, by Lemma 8. For any , there exists such that . Since , there exist such that . Since thus .
Theorem 17. Suppose, for each , and is surjective. is -complete with respect to . Then is a -basis with respect to if and only if there exists a constant such that, for any , any , and , we have
Proof. : Suppose is a -basis with respect to . Then, for any , there exists a unique sequence with for each such that . Let . Then by Theorem 12, is a norm on and it is equivalent to . So there exists a constant such that, for any , . Hence, for any , any , , we choose ; then, for any , we have : Let , and is convergent}. First, we show that . Since is -complete, is dense in . It is sufficient to show that is closed. Suppose and . Denote . Then for any and any , we have, for any , Since , so, for any , there exists , such that whenever , we have . In the above inequality, let ; we get Since is surjective, by inequality (10), we have that for any and any . So is convergent for each . Suppose . Then Since so converges to , which implies that . Thus is a closed set. Now we will show that is -linearly independent. Suppose that , where for each . Since, for any and any , we have , hence for any . But since is surjective, . So, for each . Since is arbitrary, for any . Thus is -linearly independent. So is a -basis.
4. The Property of -Minimal of -Bases
In this section, we studied the property of -minimal of -bases and the perturbations of -bases.
Theorem 18. Suppose and is surjective. Then if is a -basis, then is -minimal and -complete; if is -minimal, then is -linearly independent.
Proof. Since is a -basis, it is obvious that is -complete. We only need to prove that is -minimal. Suppose that is the unique dual -basis of . Then and are -biorthonormal by Theorem 16; that is, for any and . For any and any such that , let . Then, for any , , but , so . Hence is -minimal.
Suppose is -minimal and . Since is surjective for any , then for any and . If there exists such that , then . It follows that . Since , , which contradicts with the fact that is -minimal.
Theorem 19. Suppose that and is surjective for each . Then there exists a sequence , such that and are biorthonormal if and only if is -minimal, there exists a unique sequence such that and are biorthonormal, if and only if is -minimal and -complete.
Proof. : Suppose that is biorthonormal with ; that is, for any and . For any and any such that , let . Then, for any , , but , so . Hence is -minimal.
: Suppose that is -minimal. For all , for all such that , let . Since is surjective, . Since is -minimal, . So there exists such that and , for all . Keep in mind that the notation in statement (1) means just a general map with no linearity and bounded requirements, defining a map by if , if . Then is biorthonormal with , since for every and .
: By (1) we know that is -minimal. So it is only needed to show that is -complete. Suppose that and for any and any . Since , so , which implies that and are biorthonormal, where for each defined by for any . But it is assumed that there exists a unique sequence such that and are biorthonormal, so ; hence , which implies that . So is -complete.
: By (1), we know that if is -minimal, then there exists a sequence such that and are biorthonormal. Now we only need to show that such sequence is unique. If there is another sequence such that and are biorthonormal; then for any and . Since is -complete, by Lemma 8. Thus for every and . It follows that for every . So the sequence which is biorthonormal with is unique.
Theorem 20. Suppose that and is surjective for each . Then the following statements are equivalent: there exists a sequence which is -biorthonormal with , for all , there exists a constant such that, for any and , we have
Proof. : For any , any , let . Then for any and any , we have
Then . So satisfies the requirement.
: Let and . Suppose that and . Since so, if , then for , which follows that the vectors are uniquely defined by . Now, for each , we define operator by Then is linear and So is bounded on ; hence it can be linearly extended to . Let on ; then for each and that is, is -biorthonormal with .
5. Perturbation of -Bases
In this section, we give one result about the stability of -bases under certain perturbation.
Theorem 21. Suppose that and is surjective for each . is a -basis for with as its -dual basis and its basis constant is . If is a sequence such that , then is a -basis for with basis constant .
Proof. Suppose , . Then , since and are -biorthonormal by Theorem 16. Since which implies that if is convergent, then is convergent, the following operator is well defined: It is obvious that is linear and . It implies that is invertible. Since, for any and , we have so for any . Since is invertible, it is easy to see that is a -basis for . Now we start to show the basis constants inequality. For any , we have On the other hand, we have It follows that Since is a -basis for , by Theorem 17, we have that So, by inequalities (51), (53) and (54), we have
This work was partially supported by SWUFE’s Key Subjects Construction Items Funds 211 Projects.
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