A note on unitary similarity preserving linear mappings on \(B(H)\) (Q1765896)

The paper under review is a continuation of \textit{T. Petek} [Stud. Math. 161, No. 2, 177--186 (2004; Zbl 1065.47039)] by the same author. Let \(H\) be a complex, infinite-dimensional Hilbert space. The author studies linear surjective mappings on \({\mathcal B}(H)\) with the property that \(\Phi({\mathfrak S}_u(A))={\mathfrak S}_u(\Phi(A))\), where \({\mathfrak S}_u(X):=\{UXU^\ast\); \(U U^\ast=I=U^\ast U\}\) is the unitary orbit. It is shown that any such \(\phi\) is a unitary similarity, multiplied by a nonzero scalar \(c\) and possibly composed by transposition relative to some fixed orthonormal basis of \(H\). The proof is based on a characterization of minimal unitary invariant subspaces (i.e., the nonzero subspaces \({\mathcal V}\) of \({\mathcal B}(H)\) with the property that \({\mathfrak S}_u(X)\subseteq {\mathcal V}\) whenever \(X\in{\mathcal V}\), and which are minimal with respect to set-inclusion). It is shown that there are precisely two minimal unitary invariant subspaces, namely, \({\mathbb C}\,I\) and \(F_0(H)\), the space of finite-rank operators with zero trace.