Some unitary similarity invariant functional values preservers of matrix products

Let \({\mathcal {M}}_{n}\) ${\mathcal{M}}_n$ be the algebra of \(n\times n\) $n\times n$ complex matrices. Complete descriptions are given of the maps on \({\mathcal {M}}_{n}\) ${\mathcal{M}}_n$ leaving invariant some unitary similarity invariant functional values of the product of matrices such as the numerical radius, the pseudo spectral radius, or the condition spectral radius.