  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.
