_{1}

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.

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 [

The concept of linear 2-normed spaces has been investigated by S. Gahler in 1965 [

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.

The following definitions from [

Definition 2.1. A sequence

< 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 _{2}.

Definition 2.3. Let _{2}. Then, the analysis operator T^{*}: X ® l_{2} is the adjoint of synthesis operator T and is defined as

Definition 2.4. Let

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

Definition 2.5. X be a real linear space of dimension greater than 1 and let

a)

b)

c)

d)

Then

Definition 2.6. Let X be a linear space of dimension greater than 1 over the field K (=R or C). Suppose that

a)

b)

c)

d)

e)

Then

If

Let

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.

The definition of 2-frame from [

Definition 3.1 Let

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 [

Proposition 3.2. Let

Proof: Suppose that

Then

Similarly we can prove that

Suppose

A sequence

Definition 3.3. Let

Definition 3.4. Let

Definition 3.5. Let

Theorem 3.6. Suppose that

Proof: Since

Theorem 3.7. Suppose that

Proof: Since

Since

Therefore, the above Equation (1) is true for

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

Which shows that

Let H_{1} and H_{2} be 2-Hilbert spaces with inner products_{1} and H_{2} is denoted by

for all

where

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

Definition 4.1. Let

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

Theorem 4.2. Let _{1} and H_{2} respectively. Then, the sequence _{1} and H_{2} respectively.

Proof. Suppose that

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

This gives

That is

Therefore

where

Which shows that

Conversely, assume that

and

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

Which shows that

Hence we can have the following remark.

Remark 4.3. If the sequences

_{ }

Theorem 4.4. If

Proof. For

Hence

The following two theorems are the extension of 3.6 and 3.7 to the sequence

Theorem 4.5. Assume that

Theorem 4.6. Suppose that

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

G. Upender Reddy, (2016) Tensor Product of 2-Frames in 2-Hilbert Spaces. Advances in Pure Mathematics,06,517-522. doi: 10.4236/apm.2016.68039