a regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In this section, we redefine the notion of generalized Riesz bases, that is, $D_{\phi}$ and $D_{\psi}$ are not necessarily dense in ${\cal H}$ and show that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. [@h-t; @hiroshi1; @h-t2]. Furthermore, we have the following results:

\(i) If $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair, then for any ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$. $\bar{T}_{\phi,\bm{e}}$ (resp. $\bar{T}_{\psi,\bm{e}}$) is the minimum among constructing operators of the generalized Riesz basis $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) and $T_{\bm{e},\psi}^{-1}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the maximum among constructing operator of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$). Furthermore, any cloesd operator $T$ (resp. $K$) satisfying $\bar{T}_{\phi,\bm{e}} \subset T \subset T_{\bm{e},\psi}^{-1}$ (resp. $\bar{T}_{\psi,\bm{e}} \subset K \subset T_{\bm{e},\phi}^{-1}$) is a constructing operator for $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).

\(ii) If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the minimum (resp. 