generalized Riesz basis with a constructing pair $(\bm{e}, \bar{T}_{\phi,\bm{e}})$ (resp. $( \bm{e} , T_{\bm{e}, \phi}^{-1} )$) and the constructing operator $\bar{T}_{\phi ,\bm{e}}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).

\(3) Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then $ \{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$) is a generalized Riesz basis with a constructing pair $(\bm{e}, \bar{T}_{\psi,\bm{e}})$ (resp. $( \bm{e} , T_{\bm{e}, \psi}^{-1} )$) and the constructing operator $\bar{T}_{\psi ,\bm{e}}$ (resp. $T_{\bm{e},\psi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$).\
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[*Proof.*]{} Let $\bm{e}= \{ e_{n} \}$ be any ONB in ${\cal H}$.\
(1) Since $D(\phi)$ is dense in ${\cal H}$, it follows from Lemma 2.3 that $\bar{T}_{\phi, \bm{e}}$ has an inverse. Since $D_{\phi}$ is also dense in ${\cal H}$, it follows from Lemma 2.4 that the inverse $\bar{T}_{\phi,\bm{e}}^{-1}$ of $\bar{T}_{\phi,\bm{e}}$ is densely defined. Furthermore, since $\bar{T}_{\phi,\bm{e}}^{\ast} \psi_{n}=T_{\bm{e},\phi} \psi_{n}=e_{n}$, $n=0,1, \cdots $, we have $ \bm{e} \subset D \left( (\bar{T}_{\phi,\bm{e}}^{\ast})^{-1} \right) =D( T_{\bm{e}, \phi}^{-1})$. Thus $\{ \phi_{n} \}$ is a generalized Riesz basis with a constructing pair $( \bm{e}, \bar{T}_{\phi,\bm{e}} )$, and $\{ \psi_{n} \}$ is 