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 $(\{ ϕ_{n} \} , \{ ψ_{n} \})$ is a semi-regular biorthogonal pair, then $\{ ϕ_{n} \}$ and $\{ ψ_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. \cite{h-t, hiroshi1, h-t2} in the regular case.

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