Similarity-invariant subspaces and similarity-preserving linear maps.
Let \(\mathcal H\) be a (separable) infinite-dimensional Hilbert space, let \(\mathcal{B}(\mathcal{H})\) and \(\mathcal{K}(\mathcal{H})\) stand for the \(C^\ast\)-algebras of all bounded and compact operators, respectively, on \(\mathcal H\), and let \(I\) denote the identity operator on \(\mathcal{H}\). A closed subspace \(\mathfrak{M}\) of \(\mathcal{B}(\mathcal{H})\) is said to be similarity invariant if \(A \in \mathfrak{M}\) implies that \(T A T^{-1} \in \mathfrak{M}\) for all invertible \(T \in \mathcal{B}(\mathcal{H})\). Clearly, \(\mathcal{B}(\mathcal{H})\), \(\mathcal{K}(\mathcal{H})\), \(\mathcal{K}(\mathcal{H}) + \mathbb{C}I\), and \(\mathbb{C}I\) are similarity invariant. The authors show that these are the only ones. They then move on to characterize those bijective linear maps \(\Phi\) on \(\mathcal{B}(\mathcal{H})\) such that \(\Phi\) and \(\Phi^{-1}\) preserve similarity. Clearly, multiplication with a non-zero scalar, conjugation with an invertible operator, and forming the Banach space transpose with respect to a given orthogonal basis fall into this category. The authors show that compositions of maps of the above types are all that can arise.
scientific article; zbMATH DE number 1885848