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.