Tensor Product of 2-Frames in 2-Hilbert Spaces

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Received 9 October 2015; accepted 2 July 2016; published 5 July 2016

1. Introduction

The concept of frames in Hilbert spaces has been introduced by Duffin and Schaefer in 1952 to study some deep problems in nonharmonic Fourier series. D. Han and D.R. Larson [1] have developed a number of basic aspects of operator-theoretic approach to frame theory in Hilbert space. Peter G. Casazza [2] presented a tutorial on frame theory and he suggested the major directions of research in frame theory.

The concept of linear 2-normed spaces has been investigated by S. Gahler in 1965 [3] and has been developed extensively in different subjects by many authors. A concept which is related to a 2-normed space is 2-inner product space which has been intensively studied by many mathematicians in the last three decades. The concept of 2-frames for 2-inner product spaces was introduced by Ali Akbar Arefijammaal and Ghadir Sadeghi [4] and described some fundamental properties of them. Y. J. Cho, S. S. Dragomir, A. White and S. S. Kim [5] are presented some inequalities in 2-inner product spaces. Some results on 2-inner product spaces are described by H. Mazaherl and R. Kazemi [6] . The tensor product of frames in tensor product of Hilbert spaces is introduced by G. Upender Reddy and N. Gopal Reddy [7] and some results on tensor frame operator are presented.

In this paper, 2-frames in 2-Hilbert spaces are studied and some results on it are presented. The tensor product of 2-frames in 2-Hilbert spaces is introduced. It is shown that the tensor product of two 2-frames is a 2-frame for the tensor product of Hilbert spaces. Some results on tensor product of 2-frames are established.

2. Preliminaries

The following definitions from [2] [5] are useful in our discussion.

Definition 2.1. A sequence of vectors in a Hilbert space X is called a frame if there exist constants 0

< A ≤ B <µ such that

The above inequality is called the frame inequality. The numbers A and B are called lower and upper frame bounds respectively.

Definition 2.2. A synthesis operator T: l_{2} ®X is defined as where is an orthonormal basis for l_{2}.

Definition 2.3. Let be a frame for X and be an orthonormal basis for l_{2}. Then, the analysis operator T^{*}: X ® l_{2} is the adjoint of synthesis operator T and is defined as for all x Î X.

Definition 2.4. Let be a frame for the Hilbert space H. A frame operator is defined as for all x Î X.

Here we give the basic definitions of 2-normed spaces and 2-inner product spaces from [3] [6] .

Definition 2.5. X be a real linear space of dimension greater than 1 and let be a real-valued function on X × X satisfying the following conditions.

a) and if and only if x and y are linearly dependent vectors.

b) for all

c) for any real number and for all

d) for all

Then is called 2-norm on X and called a linear 2-normed space.

Definition 2.6. Let X be a linear space of dimension greater than 1 over the field K (=R or C). Suppose that is K-valued function on X × X × X which satisfies the following conditions.

a) and if and only if x and z are linearly dependent.

b)

c)

d)

e)

Then is called a 2-inner product on X and is called a 2-inner product space (or 2-pre Hilbert space).

If is an inner product space, then the standard 2-inner product space is defined on X by

.

Let be a 2-inner product space, we can define a 2-norm on X ´ X by, for all.

Using the above properties, we can prove the Cauchy-Schwartz inequality

A 2-inner product space X is called a 2-Hilbert space if it is complete.

3. 2-Frames

The definition of 2-frame from [1] as follows.

Definition 3.1 Let be a 2-Hilbert space and. A sequence of elements in X is called a 2-frame associated to if there exist 0 < A ≤ B <µ such that

.

The above inequality is called the 2-frame inequality. The numbers A and B are called the lower and upper 2-frame bounds respectively.

The following proposition [1] shows that in the standard 2-inner product spaces every frame is a 2-frame.

Proposition 3.2. Let be a Hilbert space and is a frame for H. Then, it is a 2-frame with the standard 2-inner product space on X.

Proof: Suppose that is a frame for X with frame bounds A and B.

Then

Similarly we can prove that. Hence is a 2-frame for 2-Hilbert space. ð

Suppose is a 2-Hilbert space and the subspace generated with a fixed element in X. Let be denote the algebraic complement of in X. So we have.We define the inner product on X as follows.

A sequence of elements in X is a 2-frame associated to with frame bounds A and B, then the definition of 2-frame can be written as.

Definition 3.3. Let be a 2-frame in X. Then, the 2-Synthesis operator is defined by.

Definition 3.4. Let be a 2-frame in X. Then, the 2-Analysis operator is defined by.

Definition 3.5. Let be a 2-frame associated to with frame bounds A and B in a 2-Hilbert space X. A 2-frame operator is defined by

Theorem 3.6. Suppose that is a sequence in 2-Hilbert space X, with holds for all if and only if is a 2-normalized tight frame for X.

Proof: Since is a 2-normalized tight frame for X, for all

for all. ð

Theorem 3.7. Suppose that is a 2-frame for Hilbert space X, and T is co-isometry. Then is a 2-frame for X.

Proof: Since is a 2-frame for X, by Definition 3.1, we have

(1)

Since is an operator, for all, we have

Therefore, the above Equation (1) is true for

By using the fact that T is co-isometry, we have

Which shows that is a 2-frame for X. ð

4. Tensor Product of 2-Frames

Let H_{1} and H_{2} be 2-Hilbert spaces with inner products, respectively. The tensor product of H_{1} and H_{2} is denoted by and is an inner product space with respect to the inner product given by

(2)

for all and. The norm on is defined by

(3)

where, and are norms generated by and respectively. The space is completion with the above inner product. Therefore, the space is a 2-Hilbert space.

The following definition is the extension of (3.1) to the sequence.

Definition 4.1. Let and be the sequences of vectors in 2-Hilbert spaces and respectively. Then, the sequence of vectors is said to be a tensor product of 2-frame for the tensor product of Hilbert spaces associated to if there exist two constants 0 < A ≤ B <µ such that

The numbers A and B are called lower and upper frame bounds of the tensor product of 2-frame, respectively.

Theorem 4.2. Let and be two sequences in Hilbert spaces H_{1} and H_{2} respectively. Then, the sequence is a tensor product of 2-frame for if and only if and are the 2-frames for H_{1} and H_{2} respectively.

Proof. Suppose that is a 2-frame for associated to. Then, for each

On using (2) and (3) the above equation becomes

This gives

That is

Therefore,

where and.

Which shows that is a 2-frame for associated to. Similarly we can prove that is a 2- frame for associated to.

Conversely, assume that is a 2-frame for associated to with frame bounds, and is a 2-frame for associated to with frame bounds,. Then

(4)

and

(5)

multiplying the Equations (4) and (5) we get

Which shows that is a tensor product of frame for. ð

Hence we can have the following remark.

Remark 4.3. If the sequences, and are the 2-frames for the Hilbert spaces, and respectively and are the frame operators respectively of above frames, then from 3.5, we have the following.

, _{ }

Theorem 4.4. If, and are the frames for the Hilbert spaces, and with the frame operators respectively, then.

Proof. For, we have

Hence. ð

The following two theorems are the extension of 3.6 and 3.7 to the sequence so, proofs are left to the reader.

Theorem 4.5. Assume that is a sequence in a Hilbert space. Then , if and only if is a 2-normalized tight frame for.

Theorem 4.6. Suppose that is a tensor product of 2-frame for, and is co-iso- metry. Then is a tensor product of 2-frame for.

Acknowledgements

The research of the author is partially supported by the UGC (India) [Letter No. F.20-4(1)/2012(BSR)].

References

[1] Han, D. and Larson, D.R. (2000) Frames, Bases and Group Representations. Memoirs of the AMS.

[2] Casazza, P.G. (2000) The Art of Frame Theory. Taiwanese Journal of Mathematics, 4, 129-201.

[3] Gahler, S. (1965) Lineare 2-Normierte Raume. Mathematische Nachrichten, 28, 1-43.

http://dx.doi.org/10.1002/mana.19640280102

[4] Arefijammaal, A.A. and Sadeghi, G. (2013) Frames in 2-Inner Product Spaces. International Journal of Mathematical Sciences and Informatics, 8, 123-130.

[5] Cho, Y.J., Dragomir, S.S., White, A. and Kim, S.S. (1999) Some Inequalities in 2-Inner Product Spaces. Demonstratio Mathematica, 32, 485-493.

[6] Mazaherl, H. and Kazemi, R. (2007) Some Results on 2-Inner Product Spaces. Novi Sad Journal of Mathematics, 37, 35-40.

[7] Reddy, G.U., Reddy, N.G. and Reddy, B.K. (2009) Frame Operator and Hilbert-Schmidt Operator in Tensor Product of Hilbert Spaces. Journal of Dynamical Systems & Geometric Theories, 7, 61-70.