Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897786 | Linear Algebra and its Applications | 2018 | 34 Pages |
Abstract
Let X and Y be two infinite-dimensional complex Banach spaces, and fix two nonzero vectors x0âX and y0âY. Let B(X) (resp. B(Y)) denote the algebra of all bounded linear operators on X (resp. on Y), and let Ex0(X) be the collection of all operators TâB(X) for which x0 is an eigenvector and (Târ1X)2 is a nonzero scalar operator for some scalar râC. We show that a map Ï from B(X) onto B(Y) satisfiesÏÏ(T)Ï(S)âÏ(S)Ï(T)(y0)=ÏTSâST(x0),(T,SâB(X)), if and only if there are two functions η:B(X)âC and ξ:B(X)â{â1,1}, and a bijective bounded linear mapping A:XâY such that Ax0=y0, the function ξ is constant on B(X)\Ex0(X), andÏ(T)=ξ(T)ATAâ1+η(T)1Y for all TâB(X).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Z. Abdelali, A. Bourhim, M. Mabrouk,