Unitary similarity invariant function preservers of skew products of operators

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$\mathcal B$besubsetsof${\mathcal B}(H)$ 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 $\mathrm{\Phi }:\mathcal{A}\to \mathcal{B}$ satisfying $F(A^{\ast }B)=F(\mathrm{\Phi }(A{)}^{\ast }\mathrm{\Phi }(B))$ 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) $F(\mu UAU^*)=F(A)$foracomplexunit$\mu$,$A\in {\mathcal F}_1(H)$andunitary$U \in {\mathcal B}(H)$(b)foranyrankoneoperator$X\in {\mathcal F}_1(H)$ the map $t\mapsto F(tX)$ on $[0,\mathrm{\infty })$ is strictly increasing, and © the set $\{F(X):X\in {\mathcal{F}}_1(H)\text{ and }\mathrm{\parallel }X\mathrm{\parallel }=1\}$ attains its maximum and minimum.