Two characterizations of automorphisms on B(X)
Volume 105 / 1993
Studia Mathematica 105 (1993), 143-149
DOI: 10.4064/sm-105-2-143-149
Abstract
Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.
