(2).\
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Next we study the relationships between the notion of biorthogonal pairs and the operators $T_{\phi,\bm{e}}$, $T_{\bm{e},\phi}$. Then we have the following statements.\

[*Lemma 2.3.*]{} [*Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ is a biorthogonal pair such that $D(\phi)$ is dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ has an inverse and $\bar{T}_{\phi,\bm{e}}^{-1} \subset T_{\bm{e},\psi}=(T_{\psi,\bm{e}})^{\ast}$.*]{}\
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[*Proof.*]{} By the definitions of $T_{\phi,\bm{e}}$ and $T_{\bm{e},\psi}$, we have $$\begin{aligned}
T_{\bm{e},\psi}T_{\phi,\bm{e}} e_{n}= T_{\bm{e},\psi} \phi_{n}=e_{n}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$ Hence we have $$\begin{aligned}
T_{\bm{e}, \psi}T_{\phi,\bm{e}}=I \;\;\; {\rm on} \;\;\; D_{\bm{e}}. \nonumber\end{aligned}$$ Thus we have $$\begin{aligned}
T_{\bm{e},\psi} \bar{T}_{\phi, \bm{e}}= I. \nonumber\end{aligned}$$ This completes the proof.\
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In general, $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is not necessarily dense in ${\cal H}$. We investigate the conditions under what $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is dense in ${\cal H}$.\

[*Lemma 2.4.*]{}

*Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ is a biorthogonal pair such that $D(\phi)$ is dense in ${\cal H}$. Then 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 $T_{\bm{e},\phi}^{-1}= ( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast}$.\
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[*Proof.*]{} (i)$\Rightarrow$(ii) Since 