The irreducibility and strong irreducibility of operators

Abstract

To construct more homogeneous operators, B. Bagchi and G. Misra in [2] introduced the operator T0 T0−T1 0 T1and proved that when T 0 and T 1 are homogeneous operators with the same unitary representation U (g), it is homogeneous with associated representation U (g) ⊕ U (g). At the same time, they asked an open question, is the constructed operator irreducible? A. Korányi in [31] showed that when the (1,2)-entry of the matrix is α(T 0 − T 1 ), α ∈ C the above result is also valid, and their unitary equivalence class depends only on |α|. In this case, he and S. Hazra [20] gave a large class of irreducible homogeneous bilateral 2 × 2 block shifts, respectively, which are mutually unitarily inequivalent for α > 0. In this note, we generalize the construction to T = T0 XT1−T0X 0 T1and provide some sufficient conditions for its irreducibility. We also find that for the above-mentioned T 0 , T 1 and non-scalar operator X, T is weakly homogeneous rather than homogeneous. So the weak homogeneity problem related to T is investigated.