Unitary similarity invariant function preservers of skew products of operators

Abstract

Let $\mathcal{B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\ge 3$, 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 $\Phi :\mathcal{A}\to \mathcal{B}$ satisfying $F(A^{\ast }B)=F(\Phi (A{)}^{\ast }\Phi (B))$ for all $A,B\in \mathcal{A}$. To establish the proofs, some general results are obtained for functions $F:{\mathcal{F}}_1(H)\cup \{0\}\to [0,+\infty )$, where ${\mathcal{F}}_1(H)$ is the set of rank one operators in $\mathcal{B}(H)$, satisfying (a) $F(\mu UA{U}^{\ast })=F(A)$ 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 $\{F(X):X\in {\mathcal{F}}_1(H)and\; \Vert X\Vert =1\}$ attains its maximum and minimum.

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