On the irreducibility and weakly homogeneity of a class of operators

Abstract

To construct more homogeneous operators, B. Bagchi and G. Misra in \cite{d} introduced the operator $\left(\begin{array}{cc}{T}_0& T_0-T_1\\ 0& T_1\end{array}\right)$ and proved that when $T_0$ and $T_1$ are homogeneous operators with the same unitary representation $U(g)$, it is homogeneous with associated representation $U(g)\oplus U(g)$. At the same time, they asked an open question, is the constructed operator irreducible? A. Kor$\acute{a}$nyi in \cite{e} showed that when the (1,2)-entry of the matrix is $\alpha (T_0-T_1)$, $\alpha \in \mathbb{C}$ the above result is also valid, and their unitary equivalence class depends only on $|\alpha |$. In this case, he and S. Hazra \cite{f} gave a large class of irreducible homogeneous bilateral $2\times 2$ block shifts, respectively, which are mutually unitarily inequivalent for $\alpha >0$. In this note, we generalize the construction to $T=\left(\begin{array}{cc}{T}_0& X{T}_1-T_{0}X\\ 0& T_1\end{array}\right)$ and 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.