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

Unitary similarity invariant function preservers of skew products of operators