operator for the ${\it quasi}$-${\it Hermitian \; operator}$ $N_{\phi,\bm{e}}$. Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then the number operator $N_{\bm{e},\psi}$ is a ${\it quasi}$-${\it Hermitian \; operator}$ for the metric operator $( \bar{T}_{\psi,\bm{e}}^{-1})^{\ast} \bar{T}_{\psi,\bm{e}}^{-1}$. The results on generalized Riesz bases is related to the problem of finding metric operators for quasi-Hermitian operators.

Semi-regular biorthogonal pairs and Psuedo-bosonic operators
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In this section, we introduce a method of constructing a semi-regular biorthogonal pair based on the following commutation rule under some assumptions. Here, the commutation rule is that a pair of operators $a$ and $b$ acting on a Hilbert space ${\cal H}$ with inner product $( \cdot | \cdot )$ satisfies $$\begin{aligned}
ab-ba=I. \nonumber\end{aligned}$$ In particular, this collapses to the canonical commutation rule (CCR) if $b= a^{\dagger}$. In Ref. [@h-t2] the author has shown assumptions to construct the regular biorthogonal pair. Indeed, the assumptions in Ref. [@h-t2] coincide with the definition of pseudo-bosons as originally given in Ref. [@bagarello10], where in the recent literature many researchers have investigated. [@bagarello13; @bagarello10; @bagarello11; @mostafazadeh; @d-t]. In this section, we introduce that some assumptions to construct the semi-regular biorthogonal pair connect with the definition of pseudo-bosons. At first, we construct 