a generalized Riesz basis with a constructing pair $(\bm{e} , T_{\bm{e}, \phi}^{-1})$. Similarly, $\{ \psi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e},\bar{T}_{\psi,\bm{e}})$, and $\{ \phi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e},T_{\bm{e},\psi}^{-1})$. Hence $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) is a generalized Riesz basis with constructing pairs $(\bm{e},\bar{T}_{\phi,\bm{e}})$ and $(\bm{e},T_{\bm{e},\psi}^{-1})$ (resp. $(\bm{e},\bar{T}_{\psi,\bm{e}})$ and $(\bm{e},T_{\bm{e},\phi}^{-1})$).

Take an arbitrary constructing operator $T$ of the generalized Riesz basis $\{ \phi_{n} \}$. Since $Te_{n}=\phi_{n}$ and $(T^{-1})^{\ast}e_{n}=\psi_{n}$, $n=0,1, \cdots $, we have $\bar{T}_{\phi,\bm{e}} \subset T$ and $\bar{T}_{\psi,\bm{e}} \subset (T^{-1})^{\ast}$, which implies that $T^{-1} \subset T_{\psi,\bm{e}}^{\ast}=T_{\bm{e},\psi}$. Hence, we have $T \subset T_{\bm{e},\psi}^{-1}$. Thus, $\bar{T}_{\phi,\bm{e}}$ and $T_{\bm{e},\psi}^{-1}$ are the minimum and the maximum among constructing operators of $\{ \phi_{n} \}$, respectively. Furthermore, suppose that $T$ is a closed operator in ${\cal H}$ such that $\bar{T}_{\phi,\bm{e}} \subset T \subset T_{\bm{e},\psi}^{-1}$. Then, since $D(T) \supset D_{\bm{e}}$, $TD(T) \supset T_{\phi,\bm{e}} D_{\bm{e}} = \{ \phi_{n} \}$ and $D((T^{\ast})^{-1}) \supset D(T_{\psi,\bm{e}}) \supset D_{\bm{e}}$, it follows that $T$ is a constructing operator for $\{ \phi_{n} \}$. Similar results for $\{ \psi_{n} \}$ are obtained.\
The statements (2) and (3) are shown similarly to (1). This completes the proof.\

[*Remark.*]{}

*Theorem 3.4 means the following:*

