\(i) $T_{\bm{e}}$ is closable.

\(ii) $T_{\phi,\bm{e}}$ is closable.

\(iii) $D( T_{\bm{e},\phi} ) = D(\phi) \equiv \left\{ x \in {\cal H} ; \sum_{k=0}^{\infty} | (x | \phi_{k})|^{2} < \infty \right\} $ is dense in ${\cal H}$.\
If this holds, then $\bar{T}_{\bm{e}}=\bar{T}_{\phi,\bm{e}}=(T_{\bm{e},\phi})^{\ast}.$\
Furthermore we investigate the relationships between the notion of biorthogonal pairs and the operators $T_{\phi,\bm{e}}$, $T_{\bm{e},\phi}$. Indeed, if $D(\phi)$ is dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ has an inverse and $\bar{T}_{\phi,\bm{e}}^{-1} \subset T_{\bm{e},\psi}=(T_{\psi,\bm{e}})^{\ast}$. However, $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is not dense in ${\cal H}$ in general. And so we may give the conditions under what $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is dense in ${\cal H}$. In detail, the following statements are equivalent:

\(i) $D_{\phi} \equiv Span \{ \phi_{n} \}$ is dense in ${\cal H}$.

\(ii) $T_{\phi,\bm{e}}$ is closable and $\bar{T}_{\phi,\bm{e}}$ has a densely defined inverse.

\(iii) $T_{\phi,\bm{e}}^{\ast}(=T_{\bm{e},\phi})$ has a densely defined inverse.\
If this holds, then $T_{\bm{e},\phi}^{-1}= ( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast}$.\
In Section 3, we first investigate the relationship between semi-regular biorthogonal pairs and generalized Riesz bases. In Definition 2.1 in Ref [@h-t], the author has defined the notion of generalized Riesz bases under the assumption that $D_{\phi}$ and $D_{\psi}$ are dense in ${\cal H}$, and has shown that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is 