General preservers of quasi-commutativity on self-adjoint operators

Abstract

Let H be a separable Hilbert space and B sa (H) the set of all bounded linear self-adjoint operators. We say that A, B ∈ B sa (H) quasi-commute if there exists a nonzero ξ ∈ C such that A B = ξ B A. Bijective maps on B sa (H) which preserve quasi-commutativity in both directions are classified.

Additive maps preserving semi-Fredholm operators with bounded ascent on $\mathcal B(\mathcal X)$

“…Linear or additive preserver problems aim to characterize linear or non-linear maps on operator algebras that preserve certain properties, subsets or relations ( [3], [8], [10], [13]- [15], [20]). One of the most famous problems in this direction is Kaplansky's problem [15] asking whether every surjective unital invertibility preserving linear map between two semi-simple Banach algebras is a Jordan homomorphism.…”