relationship between the operator $T_{\bm{e}}$ and the operators $T_{\phi, \bm{e}}$ and $T_{\bm{e},\phi}$. In Section 3, we investigate the relationship between semi-regular biorthogonal pairs and generalized Riesz bases and give the physical operators defined by the operators $T_{\phi, \bm{e}}$, $T_{\bm{e},\phi}$, $T_{\psi,\bm{e}}$ and $T_{\bm{e},\psi}$ and an ONB $\bm{e}= \{ e_{n} \}$. In Section 4, we introduce a method of constructing a semi-regular biorthogonal pair based on the pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ under some assumptions and we investigate the relationship between pseudo-bosonic operators satisfying some assumptions and the physical operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$. In Section 5, we describe future issue with respect to biorthogonal pairs $( \{ \phi_{n} \} , \{ \psi_{n} \})$ and generalized Riesz bases.

Some operators defined by biorthogonal sequences and ONB
========================================================

Let ${\cal H}$ be a Hilbert space with inner product $( \cdot | \cdot )$. We consider the following operators in ${\cal H}$ defined by a sequence $\{ \phi_{n} \}$ in a Hilbert space ${\cal H}$ and an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$: $$\begin{aligned}
T_{\phi, \bm{e}} 
&\equiv& \sum_{k=0}^{\infty} \phi_{k} \otimes \bar{e}_{k} , \nonumber \\
T_{\bm{e},\phi}
&\equiv& \sum_{k=0}^{\infty} e_{k} \otimes \bar{\phi}_{k} 