Unitary similarity invariant function preservers of skew products of operators

TL;DR

This paper characterizes surjective maps on operator subsets that preserve specific unitary invariant functions, such as pseudo spectra and numerical ranges, revealing their structure in the context of bounded linear operators on complex Hilbert spaces.

Contribution

It provides a detailed structure theorem for maps preserving various unitary invariant functions on operators, extending understanding of operator invariance properties.

Findings

Characterization of structure of maps preserving pseudo spectra and related functions

General results for functions on rank one operators satisfying invariance and monotonicity

Identification of conditions under which these functions attain maximum and minimum values

Abstract

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators. Suppose $F(\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $\Phi :{\mathcal A}\rightarrow \mathcal B$ satisfying $F(A^*B)=F(\Phi (A)^*\Phi (B))$ for all $A, B \in {\mathcal A}$. To establish the proofs, some general results are obtained for functions $F:{\mathcal F}_1(H) \cup \{0\} \rightarrow [0, +\infty)$, where ${\mathcal F}_1(H)$ is the set of rank one operators in ${\mathcal B}(H)$, satisfying (a) $F(\mu UAU^*)=F(A)$ for a complex unit $\mu$, $A\in {\mathcal F}_1(H)$ and unitary $U \in {\mathcal B}(H)$ (b) for any rank one operator $X\in {\mathcal F}_1(H)$ the map $t\mapsto F(tX)$ on $[0, \infty)$ is strictly increasing, and (c) the set $\{F(X): X \in {\mathcal F}_1(H) \hbox{ and } \|X\| = 1\}$ attains its maximum and minimum.