Subscalarity, Invariant, and Hyperinvariant Subspaces for Upper Triangular Operator Matrices

Abstract

In this paper, we introduce the class of quasi-nM-hyponormal operators and study various properties. We show that every quasi-nM-hyponormal operator is subscalar of order 2n; in particular, every M-hyponormal operator is subscalar of order two. Consequently, if the spectrum of T has a nonempty interior in \(\mathbb {C}\), then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for quasi-nM-hyponormal operators. Finally, we consider some of the spectral properties of this class and show that they share many spectral properties with normal operators on Hilbert spaces, mainly concerning Fredholm theory and local spectral theory. We are primarily interested in isolated points of the spectra of quasi-nM-hyponormal operators as well as the isolated points of the approximate point spectra. These properties lead to the concept of a polaroid-type operator, which when combined with the single-valued extension property, an important property in local spectral theory, produces a general framework for several versions of Weyl-type theorems.