Semi-regular biorthogonal pairs and generalized Riesz bases

In this paper we define the notion of semi-regular biorthogonal pairs what is a generalization of regular biorthogonal pairs in Ref. \cite{hiroshi1} and show that if $(\{{\varphi }_n\},\{{\psi }_n\})$ is a semi-regular biorthogonal pair, then $\{{\varphi }_n\}$ and $\{{\psi }_n\}$ are generalized Riesz bases. This result improves the results of Ref. \cite{h-t, hiroshi1, h-t2} in the regular case.