On the Norm of Elementary Operator

The norm of an elementary operator has been studied by many mathematicians. Varied results have been established especially on the lower bound of this norm. Here, we attempt the same problem for finite dimensional operators.


Introduction
Let H be a complex Hilbert space and ∑ N , holds remains an area of interest to many mathematicians.This paper attempts to an- swer this question for finite dimensional operators.
For a complex Hilbert space H , with dual H * , we define a finite rank operator ( ) : for all y H ∈ , where u H * ∈ , and x H ∈ is a unit vector, with: In this paper, we use finite rank operators to determine the norm of 2 E .We first review some known results on the norm of the Jordan elementary operator ( ) ( ) , : , for all ( ) fixed.We will then proceed to show that for an operator ( ) for all unit vectors x H ∈ , then: . Some mathematicians have attempted to determine the norm of n E .Timoney, used (matrix) numerical ranges and the tracial geometric mean to obtain an approximation of n E [1], while Nyamwala and Agure used the spectral resolution theorem to calculate the norm of n E induced by normal operators in a finite dimensional Hilbert space [2].
The study of the norm of the Jordan elementary operator has also attracted many researchers in operator theory.Mathieu [3], in 1990, proved that in the case of a prime C*-algebra, the lower bound of the norm of

U T S ≥
for prime JB*-algebras.
On their part, Stacho and Zalar [5], in 1996 worked on the standard operator algebra which is a sub-algebra of ( ) B H , that contains all finite rank operators.They first showed that the operator , T S U actually represents a Jordan triple structure of a C*-algebra.They also showed that if A is a standard operator algebra acting on a Hilbert space H , and , T S U T S ≥ − They later (1998), proved that , T S

U T S ≥
for the algebra of symmetric operators acting on a Hilbert space.They attached a family of Hilbert spaces to standard operator algebra, using the inner products on them to obtain their results.
In 2001, Barraa and Boumazguor [6], used the concept of the maximal numerical range and finite rank operators to show that if ( ) where, is the maximal numerical range of * T S relative to S , and * T is the Hilbert adjoint of T .Okelo and Agure [7] used the finite rank operators to determine the norm of the basic elementary operator.Their work forms the basis of the results in this paper.

The Norm of Elementary Operator
In this section, we present some of the known results on elementary operators and proceed to determine norm of the elementary operator 2 E .In the following theorem Okelo and Agure [7], determined the norm of the basic elementary operator.Theorem 2.1 [5]: Let H be a complex Hilbert space and ( ) Letting 0 ε → , we obtain: ( ) On the other hand, we have: So, setting Hence, from (1) and (2), we obtain .  For any vectors , y z H ∈ , the rank one operator, ( ) , is defined by ( ) x z y ⊗ = , for all x H ∈ .In the following three results Baraa and Boumazgour give three estimations to the lower bound of the norm of the Jordan elementary operator.See [6].Recall that the Jordan elementary operator is the operator .
We have: S S

S Ty Sy W T S y H y S T W S
. Therefore: ( ) * , sup : , and this completes the proof. Corollary 2.3: Let H be a complex Hilbert space and , T S be bounded linear operators on H . Let Recall that in the previous theorem (Inequality (3)), we obtained:  For each 1 n ≥ , we have: So, for all 0 ε > , ( ) ( ) Therefore, ( ) Next, we show that ( ) the set of bounded operators on H .A basic elementary operator, a finite sum of the basic elementary operators, de-

Theorem 2
is the maximal numerical range of * T S relative to S , as defined earlier.
we can find two sequences { } 1 n n x ≥ and { } 1 n n y ≥ of unit vectors in H such that: completes the proof. Let H be a complex Hilbert space and , T S be bounded linear operators on H .If