\(1) Suppose $D(\phi)$ and $D_{\phi}$ (resp. $D(\psi)$ and $D_{\psi}$) are dense in ${\cal H}$. Even if $D_{\psi}$ (resp. $D_{\phi}$) is not dense in ${\cal H}$, $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$) becomes a generalized Riesz basis.

\(2) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, but $D_{\psi}$ is not dense in ${\cal H}$. As shown in Theorem 3.4, $\bar{T}_{\phi,\bm{e}}$ is the minimum among constructing operators of $\{ \phi_{n} \}$, however the maximal constructing operator of $\{ \phi_{n} \}$ does not necessarily exist because $T_{\bm{e},\psi}^{-1}$ is not a constructing operator of $\{ \phi_{n} \}$ different to the case of regular biorthogonal pair. Furthermore, $T_{\bm{e},\phi}^{-1}$ is the maximum among constructing operators of $\{ \psi_{n} \}$, however the minimal constructing operator of $\{ \psi_{n} \}$ does not necessarily exist because $\bar{T}_{\psi,\bm{e}}$ is not a constructing operator of $\{ \psi_{n} \}$. Similar results for the case that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, but $D_{\phi}$ is not dense in ${\cal H}$ are obtained.

\
\
By Theorem 3.4, Ref. [@h-t] and [@h-t2], we can define the physical operators as follows:\

\(1) Suppose $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then, we put $$\begin{aligned}
A_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} 