Jordan product and analytic core preservers

TL;DR

This paper characterizes the structure of surjective maps on the algebra of bounded linear operators on an infinite-dimensional Banach space that preserve the analytic core of symmetrized products.

Contribution

It determines the explicit form of surjective maps preserving the analytic core of symmetrized operator products on infinite-dimensional Banach spaces.

Findings

Identifies the form of maps preserving the analytic core of symmetrized products.

Provides conditions under which such maps are characterized.

Advances understanding of operator algebra preservers.

Abstract

Let $\mathcal{B} (X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. For an operator $ T \in \mathcal{B} (X)$, $K(T)$ denotes as usual the analytic core of $T$. We determine the form of surjective maps $ \phi $ on $ \mathcal{B} (X)$ satisfying $$ K(\phi(T) \phi(S) + \phi (S) \phi (T)) = K(TS + ST) $$ for all $T, S \in \mathcal{B} (X)$.