标题： 一类算子的不可约性和弱齐次性 显示英文标题

标题： On the irreducibility and weakly homogeneity of a class of operators

摘要： 为了构造更均匀的算子，B. Bagchi 和 G. Misra 在\cite{d}中引入了算子$\left(\begin{smallmatrix} T_0 & T_0-T_1 \\ 0 & T_1\\ \end{smallmatrix}\right)$并证明了当$T_0$和$T_1$是具有相同酉表示$U(g)$的均匀算子时，它与关联表示$U(g)\oplus U(g)$也是均匀的。 与此同时，他们提出了一个开放性问题，构造的算子是否是不可约的？ A. Kor$\acute{a}$nyi in \cite{e}表明当矩阵的(1,2)项为$\alpha(T_0-T_1)$， $\alpha\in\mathbb{C}$时，上述结果也成立，且它们的酉等价类仅取决于$|\alpha|$。 In this case, he and S. Hazra \cite{f} gave a large class of irreducible homogeneous bilateral $2\times2$ block shifts, respectively, which are mutually unitarily inequivalent for $\alpha>0$. In this note, we generalize the construction to $T=\left(\begin{smallmatrix} T_0 & XT_1-T_0X \\ 0 & T_1\\ \end{smallmatrix}\right)$ 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. 因此，与$T$相关的弱齐性问题被进行了研究。 显示英文摘要

摘要： To construct more homogeneous operators, B. Bagchi and G. Misra in \cite{d} introduced the operator $\left(\begin{smallmatrix} T_0 & T_0-T_1 \\ 0 & T_1\\ \end{smallmatrix}\right)$ 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 $2\times2$ block shifts, respectively, which are mutually unitarily inequivalent for $\alpha>0$. In this note, we generalize the construction to $T=\left(\begin{smallmatrix} T_0 & XT_1-T_0X \\ 0 & T_1\\ \end{smallmatrix}\right)$ 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.