关于初等算子的本性范数

On the Essential Norm of Elementary Operators

本文讨论了两个特殊的初等算子的本性范数.它们是初等乘法算子r=r(A,B):T→ATB以及广义导算子τ=τ(A,B):T→AT—TB.主要结果是(定理2.5,定理5.2):为了给出后一表达式,我们引进了算子本性规一极大值域(§3)以及两算子的中心、本性中心和混合中心,(§4)等概念;讨论了它们的性质和相互联系.最后,在§6中,我们给出了一个使一般初等算子△的本性范数的充要条件.

The essential norm of two special elementary operators is discussed in this paper.They are the elementary multiplication operator г = г (A, B): T→ATB, and the generalizedderivation τ=τ(A,B):T→AT-TB. The main results are (theorem 2. 5 and theorem 5. 2):||г||e=max{||A||e||||B||;e}||τ||e=max{infλ∈c[||A-λ||e+||B-λ||;infλ∈c||A-λ||+||-λ||e]}In order to prove the later expression,we introduce the concepts such as essential normal-ized maximal range for an operator(§3),and center,essential center and mixed center of twooperators(§4). Their properties and relationships are discussed in the paper.Finally, in section 6, we point out a sufficient and necessary condition for the essentialnorm of the elementary operator △: