\phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In detail, we have the following\

[*Theorem 3.4.*]{}

*Let $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ be biorthogonal sequences in ${\cal H}$, and let $\bm{e}= \{ e_{n} \}$ be an arbitrary ONB in ${\cal H}$. Then the following statements hold:*

\(1) Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair. Then $\{ \phi_{n} \}$ (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 $(\bm{e}, T_{\bm{e},\phi}^{-1})$), and $\bar{T}_{\phi,\bm{e}}$ (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 maximal among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$). Furthermore, any closed 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. $\{ \psi_{n} \}$).

\(2) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then $ \{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) is a 