Tensor Product of 2-Frames in 2-Hilbert Spaces

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.


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.

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

Te x =
where { } i e is an orthonormal basis for l 2 .
be a frame for X and { } i e 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 Here we give the basic definitions of 2-normed spaces and 2-inner product spaces from [3]  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) ( ) , 0 x x z ≥ and ( ) if and only if x and z are linearly dependent.

If ( )
, X is an inner product space, then the standard 2-inner product space ( ) .,. . is defined on X by ( ) , , , , .,. .X be a 2-inner product space, we can define a 2-norm on X × X by , , x y x x y = , for all , x y X ∈ .
Using the above properties, we can prove the Cauchy-Schwartz inequality ( )
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.
is a frame for H.Then, it is a 2-frame with the standard 2-inner product space on X.
Similarly we can prove that ( ) , .,. .X is a 2-Hilbert space and L ξ the subspace generated with a fixed element ξ in X.Let M ξ be denote the algebraic complement of L ξ in X.So we have L M X ξ ξ ⊕ = .We define the inner product .,. ξ on X as follows , , of elements in X is a 2-frame associated to ξ with frame bounds A and B, then the defi- nition of 2-frame can be written as ( ) , for all be a 2-frame in X.Then, the 2-Synthesis operator be a 2-frame in X.Then, the 2-Analysis operator be a 2-frame associated to ξ with frame bounds A and B in a 2-Hilbert space X.
is a sequence in 2-Hilbert space X, with ( ) is a 2-normalized tight frame for X.
, , , , is a 2-frame for Hilbert space X, and T is co-isometry.Then { } is a 2-frame for X, by Definition 3.1, we have Since : T X X * → is an operator, for all x H ∈ , we have T x X * ∈ Therefore, the above Equation ( 1) is true for T x X * ∈ ( ) , for all By using the fact that T is co-isometry, we have ( ) , for all is a 2-frame for X.

H H ⊗
and is an inner product space with respect to the inner product given by ( ) ( ) ( ) for all 1 The numbers A and B are called lower and upper frame bounds of the tensor product of 2-frame, respectively.
On using ( 2) and (3) the above equation becomes multiplying the Equations ( 4) and ( 5) we get ( ) x y H H ⊗ ∈ ⊗ , we have The following two theorems are the extension of 3.6 and 3.7 to the sequence { } i j x y ⊗ so, proofs are left to the reader.