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.