Multiplicatively local spectrum-preserving maps

作者：Bourhim Abdellatif^*; Lee Ji Eun

来源：Linear Algebra and Its Applications, 2018, 549: 291-308.

DOI：10.1016/j.laa.2018.03.042

• Nonlinear preservers
• Local spectrum
• The single-valued extension property
• Finite rank operators

Let X and Y be two infinite-dimensional complex Banach spaces, and B(X) (resp. B(Y)) be the algebra of all bounded linear operators on X (resp. on Y). Fix two nonzero vectors x(0) is an element of X and y(0) is an element of Y, and let B-x0 (X)(resp. B-v0(Y)) be the collection of all operators in B(X) (resp. in B(Y)) vanishing at x(0) (resp. at y(0)). We show that if two maps phi(1) and phi(2) from B(X) onto B(Y) satisfy
sigma(phi 1)(S)(phi 2),(T)(y0) = sigma(ST)(x(0)), (S, T is an element of B(X))
then cpa maps B-x0(X) onto B-y0(Y) and there exist two bijective linear mappings A : X -> Y and B : Y -> X such that Ax(0)= y(0), and phi(1)(T) = ATB for all T is an element of B(X) and phi(2)(T) = B(-1)TA(-1) for all T is not an element of B-x0(X). When X = Y = C-n, we show that the surjectivity condition on phi(1) and phi(2) is redundant. Furthermore, some known results are obtained as immediate consequences of our main results.

• 出版日期2018-7-15