\sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \\
B_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \\
N_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \end{aligned}$$ $$\begin{aligned}
A_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) T_{\bm{e}, \phi}, \nonumber \\
B_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \phi}, \nonumber \\
N_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \phi}. \nonumber \end{aligned}$$

\(2) Suppose $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then, we put $$\begin{aligned}
A_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
B_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
N_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
A_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) T_{\bm{e}, \psi}, \nonumber \\
B_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \psi}, \nonumber \\
N_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \psi}. \nonumber \end{aligned}$$ Then we have the following\

[*Theorem 3.5.*]{}

*The following statements hold.*

\(1) Suppose that $D(\phi)$ and 