Hyperinvariant Subspace Problem for Some Classes of Operators

Abstract

An n-normal operator may be defined as an $n \times n$ operator matrix with entries that are mutually commuting normal operators and an operator $T \in \mathcal {B(H)}$ is quasi-nM-hyponormal (for $n \in \mathbb {N}$) if it is unitarily equivalent to an $n \times n$ upper triangular operator matrix $(T_{ij})$ acting on $\mathcal {K}^{(n)}$, where $\mathcal {K}$ is a separable complex Hilbert space and the diagonal entries $T_{jj}$ $(j = 1,2,\ldots , n)$ are M-hyponormal operators in $\mathcal {B(K)}$. This is an extended notion of n-normal operators. We prove a necessary and sufficient condition for an $n \times n$ triangular operator matrix to have Bishop’s property $(\beta )$. This leads us to study the hyperinvariant subspace problem for an $n \times n$ triangular operator matrix.