On Norm of Elementary Operator: An Application of Stampfli’s Maximal Numerical Range

Many researchers in operator theory have attempted to determine the relationship between the norm of an elementary operator of finite length and the norms of its coefficient operators. Various results have been obtained using varied approaches. In this paper, we attempt this problem by the use of the Stampfli’s maximal numerical range.


Introduction
Properties of elementary operators have been investigated in the resent past under varied aspects. Their norms have been a subject of interest for research in operator theory. Deriving a formula to express the norm of an arbitrary elementary operator in terms of its coefficient operators remains a topic of research in operator theory. In the current paper, the concept of the maximal numerical range is applied in determining the lower bound of the norm of an elementary operator consisting of two terms, and also to determine the conditions under which the norm of this operator is expressible in terms of its coefficient operators in . Specifically, the Stampfli's maximal numerical range is employed in arriving at our results.
Let be a complex Hilbert space and be the set of bounded linear operators on . We define an elementary operator, Let # be an algebra. A derivation is a function Δ: # ⟶ # for which Δ %, & % Δ& Δx y for all %, & ∈ # . If there is an ) ∈ # such that ∆% %) + )% for all % ∈ # , then )? ? is called an inner derivation. A derivation is another example of elementary operators.
Given and elementary operator on with fixed operators , on for ! 1,2 , does the relationship ‖ ‖ ∑ ‖ ‖‖ ‖ hold? King'ang'i [1] attempted this problem using the maximal numerical range of * relative to . The current paper employs the concept of the Stampfli's maximal numerical range to determine the lower bound of the norm of elementary operator , and also to determine the conditions under which the norm of this operator is expressible in terms of the norms of its coefficient operators in . The approach used by Barraa and Boumazguor [2], and also by King'ang'i [1] is employed in obtaining our results.

The Norm of the Jordan Elementary Operator
Let be a complex Hilbert space, be the algebra of bounded linear operators on , and , ∈ be fixed. For a Jordan elementary operator " , , Mathieu [5], in 1990 proved that in the case of prime C*-algebras, the lower bound of the norm of " , can be estimated by In 1994, Cabrera and Rodriguez [3], proved that for prime JB*-algebras. On their part, Stacho and Zalar [6], in 1996, worked on the standard operator algebra (which is a sub-algebra of that contains all finite rank operators). They first showed that the operator " , actually represents a Jordan triple structure of a C*-algebra. They also showed that if C is a standard operator algebra acting on a Hilbert space , and , ∈ C, then >" , > ≥ 2 √2 − 1 ‖ ‖‖ ‖.
They later (1998), proved that >" , > ≥ ‖ ‖‖ ‖ for the algebra of symmetric operators acting on a Hilbert space. They attached a family of Hilbert spaces to standard operator algebra and used the inner products in them to obtain their results. Barraa and Boumazguor [2], in the year 2001 used the concept of the numerical range of relative to , denoted by * , to obtain their results. They employed the idea of finite rank operators to prove the following theorem;

Norm of Elementary Operator of Length Two
Kingangi et al [4] in 2014 used finite rank operators to determine the norm of the elementary operator . Below is the theorem they proved (see theorem 2.5): For the proof of theorem 3.1, see King' ang'i [4], theorem 2.5.
Jocic et al [8], proved that if . V  and ∆ by analogy. Wafula, et al [9], considered normally represented elementary operators. They proved that the norm of an elementary operator is equal to the largest singular value of the operator itself. They also proved that, if the Jordan elementary operator " , , we have; >" , > j ≥ 2√2 − k‖ ‖‖ ‖, where , ∈ . In 2017, King'ang'i [1] employed the concept of the maximal numerical range of * relative to to determine the lower norm of an elementary operator of length two. He proved the following theorem (see theorem 3. , then,‖ ‖ = ∑ ‖ ‖‖ ‖. Below, we present more results on the norm of this operator by employing the concept of theStampfli's maximal numerical range. In theorem 3.5, we determine the lower bound of the norm of the operator while in theorem 3.6 we determine the conditions necessary to express the norm of in the form ‖ ‖ = ∑ ‖ ‖‖ ‖. Theorem 3.5 Let be an elementary operator on and , ∈ . If / ∈for each / ∈ ℂ, ! = 1,2, then we have ‖ ‖ ≥ sup I q ∈J r q .‖∑ / ‖: ∈ , ! = 1,24.
In the next theorem, the condition necessary for the norm of the elementary operator to be equal to the sum of the product of the norms of the corresponding coefficient operators in its definition is given. Therefore, taking limits as → ∞, we obtain; >∑ q , q > ≥ ‖ ‖ ‖ ‖ + ‖ ‖ ‖ ‖ + 2‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ .

Conclusions
In this paper, we have determined the lower bound of the norm of an elementary operator of length two in a C*algebra and using the Stampfli's maximal numerical range. The conditions in which this norm is equal to the sum of the products of the corresponding coefficient operators has also been considered. One may attempt this problem for an elementary operator consisting of more that two terms.