\nonumber \\
B_{\bm{e},\psi} \phi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \phi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
N_{\bm{e},\psi} \psi_{n}
&=& n \psi_{n} . \nonumber\end{aligned}$$ Hence $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ are lowering, raising and number operators for $\{ \psi_{n} \}$, respectively, and $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ are raising, lowering and number operators for $\{ \phi_{n} \}$, respectively.\

[*Remark.*]{}

**

\(i) In case of (1), since $$\begin{aligned}
A_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \\
B_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \\
N_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \end{aligned}$$ the author has denoted $A_{\bm{e}}^{\dagger}$, $B_{\bm{e}}^{\dagger}$ and $N_{\bm{e}}^{\dagger}$ in Ref. [@h-t2].

\(ii) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then the number operators $N_{\phi,\bm{e}}$ and $N_{\bm{e},\phi}( \equiv N_{\phi,\bm{e}}^{\dagger})$ for $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$, respectively have the relation: $( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1} N_{\phi,\bm{e}}= N_{\phi,\bm{e}}^{\dagger} (\bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1}$. This is called that $N_{\phi,\bm{e}}$ is a ${\it quasi}$-${\it Hermitian \; operator}$ [@s-k2012; @s-g-h1992; @j-d1961] and positive self-adjoint operator $( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1}$ is often called a metric 