x= Tx$. Hence $D(T^{\ast}) \cap R(T)$ is dense in $R(T)$, and since $R(T)$ is dense in ${\cal H}$, it follows from (2.1) that $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$. This completes the proof.\
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In Ref. [@hiroshi1], we have shown that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In order to generalize this result, we define the notion of semi-regular biorthogonal pair as follows:\

[*Definition 3.3.*]{} [*A pair $( \{ \phi_{n} \} , \{ \psi_{n} \})$ of biorthogonal sequences in ${\cal H}$ is said to be semi-regular if either $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ or $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$.*]{}\
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We give a concrete example[@bagarello13] of semi-regular and non regular biorthogonal bases. Let $\{ e_{n} \}$ be an ONB in ${\cal H}$ and put $\phi_{n}=e_{n}+e_{0}$ and $\psi_{n}=e_{n}$, $n=1,2, \cdots$. Then it is easily shown that $\{ \phi_{n} \}$ and $ \{ \psi_{n} \}$ are biorthogonal bases such that $D_{\phi}$ and $D(\phi)$ are dense in ${\cal H}$, but $D_{\psi}$ is not dense in ${\cal H}$. We show that if $( \{ 