Semi-regular biorthogonal pairs and generalized Riesz bases

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