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abstract: |
    In this paper we define the notion of semi-regular biorthogonal pairs what is a generalization of regular biorthogonal pairs in Ref. [@hiroshi1] and show that if $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. [@h-t; @hiroshi1; @h-t2] in the regular case.\
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[**Semi-regular biorthogonal pairs and generalized Riesz bases**]{}

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Hiroshi Inoue\

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Introduction
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Let ${\cal H}$ be a Hilbert space with inner product $( \cdot | \cdot )$, $\bm{e}= \{ e_{n} \}$ an ONB in ${\cal H}$ and $\{ \phi_{n} \}$ a sequence in ${\cal H}$. In Ref. [@hiroshi1], the author has defined an operator $T_{\bm{e}}$ on $D_{\bm{e}} \equiv Span \{ e_{n} \}$ by $$\begin{aligned}
T_{\bm{e}} \left( \sum_{k=0}^{n} \alpha_{k}e_{k} \right)
= \sum_{k=0}^{n} \alpha_{k} \phi_{k} . \nonumber\end{aligned}$$ By using this operator $T_{\bm{e}}$, the author has investigated the relationship between a regular biorthogonal pair $( \{ \phi_{n} \} , \{ \psi_{n} \})$ and the notions of Riesz bases and semi-Riesz bases. Here, $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a pair of Riesz bases if there exists an ONB $\bm{e}= \{ e_{n} 