LAUSR

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