Norm for Sums of Two Basic Elementary Operators

We give necessary and sufficient conditions under which the norm of basic elementary operators attains its optimal value in terms of the numerical range. 1. Introduction Let be a normed space over ( or ), its unit sphere, and its dual topological space. Let be the normalized duality mapping form to given by Let be the normed space of all bounded linear operators acting on . For any operator and , is called the spatial numerical range of , which may be defined as This definition was extended to arbitrary elements of a normed algebra by Bonsall [1–3] who defined the numerical range of as where is the left regular representation of in , that is, for all . is known as the algebra numerical range of , and, according to the above definitions, is defined by For an operator , Bachir and Segres [4] have extended the usual definitions of numerical range from one operator to two operators in different ways as follows. The spatial numerical range of relative to is The spatial numerical range of relative to is The maximal spatial numerical range of relative to is For , let , then the set is called the generalized maximal numerical range of relative to . It is known that is a nonempty closed subset of and . The definition of can be rewritten, with respect to the semi-inner product as with respect to an inner product as We shall be concerned to estimate the norm of the elementary operator , where are bounded linear operators on a normed space and is the basic elementary operator defined on by We also give necessary and sufficient conditions on the operators under which attaints its optimal value . 2. Equality of Norms Our next aim is to give necessary and sufficient conditions on the set of operators for which the norm of equals . Lemma 2.1. For any of the operators and all , one has Proof. The proof is elementary. Theorem 2.2. Let be operators in . If and , then Proof. The proof will be done in four steps; we choose one and the others will be proved similarly. Suppose that and , then there exist such that and there exist such that . Define the operators as follows: Then , and Hence Letting , Since therefore Corollary 2.3. Let be a normed space and . Then, the following assertions hold: (1)if , then ; (2)if and , then . Remark 2.4. In the previous corollary, if we set , then we obtain an important equation called the Daugavet equation: It is well known that every compact operator on [5] or on [6] satisfies (2.10). A Banach space is said to have the Daugavet property if every rank-one operator on satisfies (2.10). So that from our Corollary 2.3 if or for every rank-one