Quasireducible operators (Q1811961)

Let \({\mathcal H}\) be a complex Hilbert space of dimension greater than one. The author introduces the concept of quasireducible operator, generalizing the standard characteristic property of reducible operators. A bounded linear operator \(T\) on \({\mathcal H}\) is quasireducible, if there exists a nonscalar operator \(L\) on \({\mathcal H}\) such that \[ LT=TL, \quad \text{rank}((T^{\ast}L-LT^{\ast})T-T(T^{\ast}L-LT^{\ast}))\leq 1. \] There are given some properties of these operators, e.g.: a quasireducible essentially normal operator has a nontrivial invariant subspace.