Semi-regular biorthogonal pairs and generalized Riesz bases

TL;DR

This paper introduces semi-regular biorthogonal pairs, generalizing regular biorthogonal pairs, and proves that such pairs form generalized Riesz bases, extending previous results in the regular case.

Contribution

It defines semi-regular biorthogonal pairs and demonstrates they are generalized Riesz bases, advancing the theory beyond regular biorthogonal pairs.

Findings

Semi-regular biorthogonal pairs are generalized Riesz bases.

The result extends previous work on regular biorthogonal pairs.

Improves understanding of biorthogonal pairs in functional analysis.

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.