$A_{\bm{e},\psi}$ and $B_{\bm{e},\psi}$). The proofs are easily shown.\

[*Proposition 4.3.*]{}

*If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then the following statements hold.*

\(1) $$\begin{aligned}
\phi_{n}&=& \frac{1}{\sqrt{n !}} B_{\phi, \bm{e}}^{n} \phi_{0}, \;\;\; n=0,1, \cdots , \nonumber \\
\psi_{n}&=& \frac{1}{\sqrt{n!}} A_{\bm{e}, \phi}^{n} \psi_{0}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$

\(2) $$\begin{aligned}
A_{\phi, \bm{e}} D_{\phi}
= D_{\phi}, && B_{\phi, \bm{e}}D_{\phi} =D_{\phi} , \nonumber \\
& {\rm and}& \nonumber \\
A_{\bm{e}, \phi} D_{\psi}
= D_{\psi}, &&
B_{\bm{e}, \phi} D_{\psi}
=D_{\psi} . \nonumber\end{aligned}$$

[*Proposition 4.4.*]{}

*If $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then the following statements hold.*

\(1) $$\begin{aligned}
\psi_{n}&=& \frac{1}{\sqrt{n !}} B_{\psi, \bm{e}}^{n} \psi_{0}, \;\;\; n=0,1, \cdots , \nonumber \\
\phi_{n}&=& \frac{1}{\sqrt{n!}} A_{\bm{e}, \psi}^{n} \phi_{0}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$

\(2) $$\begin{aligned}
A_{\psi, \bm{e}} D_{\psi}
= D_{\psi}, && B_{\psi, \bm{e}}D_{\psi} =D_{\psi} , \nonumber \\
& {\rm and}& \nonumber \\
A_{\bm{e}, \psi} D_{\phi}
= D_{\phi}, &&
B_{\bm{e}, \psi} D_{\phi}
=D_{\phi} . \nonumber\end{aligned}$$

Next we investigate the relationship between pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ satisfying Assumption 1-3 and the operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$, $A_{\bm{e},\phi}$ and $B_{\bm{e},\phi}$ ($A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$, $A_{\bm{e},\psi}$ and $B_{\bm{e},\psi}$).\
By Proposition 4.1, Proposition 4.2 and Theorem 3.5 we have the following\

[*Lemma 4.5.*]{}

*The following statements hold.*

\(1) If $D(\phi)$ and $D_{\phi}$ 