the maximum) among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).

\(iii) If $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then $\bar{T}_{\psi,\bm{e}}$ (resp. $T_{\bm{e},\psi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$).\
We study 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 $\bm{e}= \{ e_{n} \}$. If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then lowering, raising and number operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ for $\{ \phi_{n} \}$ are defined, respectively, and raising, lowering and number operators $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ for $\{ \psi_{n} \}$ are defined, respectively. Furthermore, if $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then lowering, raising and number operators $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ for $\{ \psi_{n} \}$ are defined, respectively, and raising, lowering and number operators $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ for $\{ \phi_{n} \}$ are defined, respectively. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. [@mostafazadeh; @bagarello11; @bagarello2013] Many researchers have investigated such operators mathematically. [@h-t; @h-t2; @hiroshi1; @b-i-t]

In Section 4, we shall show a method of constructing a semi-regular biorthogonal pair based on the following commutation 