Reducibility of Invariant Subspaces of Operators Related to k-Quasiclass-A(n) Operators

Abstract

It is well-known that hyponormal operators have many interesting properties, for example, if the restriction \(T|_{{\mathcal {M}}}\) of the hyponormal operator T on its nontrivial closed invariant subspace \({\mathcal {M}}\) is normal, then \({\mathcal {M}}\) reduces T. In order to discuss the reducibility of invariant subspaces of an operator, four properties of invariant subspaces (\(R_{i}, i=1,\ldots ,4\)) are introduced. Among others, it is proved that, for a k-quasi-A(n) operator T, if the restriction \(T|_{{\mathcal {M}}}\) is normal and injective, then \({\mathcal {M}}\) reduces T, thus the function \(\sigma :T\longmapsto \sigma (T)\) is continuous on the class of k-quasiclass-A(n) operators. Some examples related to class A(n) and n-paranormal operator are given which imply that the inclusion relations are strict.