NUMERICAL RADIUS INEQUALITIES IN 2-INNER PRODUCT SPACES

. In this paper, we have obtained the analogue results on numerical radius inequalities from the classical inner product spaces to 2-inner product spaces. We have established several related reverse inequalities and some well known results in 2-inner product spaces.

Using the above properties, one has proved that Cauchy-Schwartz inequality (see [5]) It should be noticed that, the most standard example for a linear 2-inner product (·, ·|·) is defined on X by for all x, y, z ∈ X . In [2], it is shown that, in any given 2-inner product space (X , (·, ·|·)), we can define a function for all x, z ∈ X . It is not hard to see that this function satisfies the following conditions (see [6]): (N1) x, y = 0 if and only if x and y are linearly dependent; (N2) x, y = y, x ; (N3) αx, y = |α| x, y for any real number α; Any function ·, · defined on X × X and satisfying the above conditions is called a 2-norm induced from a 2-inner product on X and (X , ·, · ) is called linear 2-normed space. Some of the basic properties of 2-norms are that they are non-negative and x, y + αx = x, y , for all x, y ∈ X and all α ∈ R. Whenever a 2-inner product space (X , (·, ·|·)) is given, we consider it as a linear 2-normed space (X , ·, · ) with the 2-norm defined by (1.2).
An operator A ∈ B (X ) is said to be bounded if there exists a real number M > 0 such that for every x, y ∈ X . The norm of the b-operator is defined by [9]: where b is fixed element in X . We can easily verify that the left-hand side of ( Harikrishnan et al. in [8] proved the Riesz theorem in 2-inner product spaces. As a consequence of their work, we have for each x, y ∈ X and fixed element b ∈ X .
Recently, M. E. Omidvar et al. [10] established various reverses of the Cauchy-Schwarz and triangle inequalities in 2-inner product spaces.
In this paper, we introduce the concepts of b-numerical radius in 2-inner product spaces. Some fundamental inequalities related to the b-numerical radius of bounded linear operators in 2-inner product spaces are established.

Main Results
We first review some basic facts about numerical range and numerical radius in Hilbert space H , then try to define them in a 2-inner product space. Let (H , ·, · ) be a complex Hilbert space and B (H ) denote the C * -algebra of all bounded linear operators on H . An operator The following properties of W (A) are immediate: The most important classical fact about the geometry of the numerical range is that it is convex and its closure contains the spectrum of the operator. The usual operator norm of A, is defined by It is well known that ω (·) defines a norm on B (H ) and that for every A ∈ B (H ), we have Thus, the usual operator norm and the numerical radius norm are equivalent. See [7] for a discussion and further references. Now we are in a position to state the main result of this section. The b-numerical range of A ∈ B (X ), denoted by W b (A), is the subset of the complex numbers given by The b-numerical radius of A ∈ B (X ), denoted by ω b (A), is defined by It is easy to see that, for any (x, b) ∈ X × b , we have The b-numerical radius ω b (A) of an operator A on X is a norm on B (X ) , this norm is equivalent to the b-operator norm. In order to get our main result, we need the following lemmas: for any x, y, z ∈ X .

Lemma 2.2 ([4]).
For every x, y ∈ X , we have We shall, however, present another result, which is a possible generalization of (2.1).

Proposition 2.1. For each
On the other hand, by Lemma 2.1 and Lemma 2.2 we get By taking supremum over x, b = y, b = 1, we deduce the desired result.
Theorem 2.1. Let A, B ∈ B (X ) and AB = BA, then Proof. We may assume ω b (A) = ω b (B) = 1 and show that ω b (AB) ≤ 2. By the triangle inequality, the power inequality theorem, and the subadditivity of ω (·), we have The following simple result provides a connection between the numerical radius and b-numerical radius as follows: and b ∈ X is a fixed element.
Proof. We observe that By taking supremum over x, b ≤ 1 we deduce the desired result (2.2).
The following inequalities may be stated as well.
where I is the identity operator on X , then Taking supremum over (x, b) ∈ X , b , with x, b = 1 we get the following inequality Proposition 2.2. Let A ∈ B (X ) be a non zero bounded linear operator on the linear 2-normed space X and λ ∈ C\ {0} and α > 0 with |λ| > α. If Proof. From (2.6) of Theorem 2.3, we have and by (2.9) we deduce which is equivalent to (2.8). Proof. From Proposition 2.1, we infer that 1

Corollary 2.3. Let
. Combining the above two inequalities one can