$\psi_{n} \equiv (T^{-1})^{\ast} e_{n}$, $n=0,1, \cdots$. Then, $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are biorthogonal and $(T^{-1})^{\ast}$ is a densely defined closed operator in ${\cal H}$ with densely defined inverse $T^{\ast}$. Hence $\{ \psi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e} , (T^{-1})^{\ast} )$.

\(3) $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$.\

[*Proof.*]{} (1) and (2) are easily shown.\
(3) We first show that $$D(T^{\ast}) \subset D(\phi)  \;\;\; {\rm and} \;\;\; R(T)=D(T^{-1}) \subset D(\psi). \tag{2.1}$$ Indeed, this follows from $$\begin{aligned}
\sum_{k=0}^{\infty} |(x| \phi_{k})|^{2}
&=& \sum_{k=0}^{\infty} |(T^{\ast}x |e_{k})|^{2} \nonumber \\
&=& \| T^{\ast}x \|^{2}, \;\;\; x \in D(T^{\ast}) \nonumber\end{aligned}$$ and $$\begin{aligned}
\sum_{k=0}^{\infty} |( y| \psi_{k} )|^{2}
&=& \sum_{k=0}^{\infty} |(T^{-1}y |e_{k})|^{2} \nonumber \\
&=& \| T^{-1} y \|^{2}, \;\;\; y \in D(T^{-1}). \nonumber\end{aligned}$$ Since $D(T^{\ast})$ and $R(T)$ are dense in ${\cal H}$, it follows that $D(\phi)$ and $D(\psi)$ are dense in ${\cal H}$. Next we show that $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$. Take an arbitrary $x \in D(T)$. Let $|T|= \int_{0}^{\infty} \lambda d E_{T}( \lambda)$ be the spectral resolution of the absolute $|T| \equiv (T^{\ast}T)^{\frac{1}{2}}$ of $T$. Then we have $TE_{T}(n) x \in D(T^{\ast}) \cap R(T)$, $n=0,1, \cdots$ and $\lim_{n \rightarrow \infty} TE_{T}(n) 