# Generalized Riesz systems and quasi bases in Hilbert space

## TL;DR

This paper introduces the concept of $(D, E)$-quasi basis in Hilbert spaces and demonstrates that such biorthogonal sequences are generalized Riesz systems, which are useful in constructing non-self-adjoint Hamiltonians.

## Contribution

It defines $(D, E)$-quasi bases and proves their sequences are generalized Riesz systems, linking them to physical operator constructions.

## Key findings

- $(D, E)$-quasi bases are introduced for dense subspaces.
- Biorthogonal sequences forming a $(D, E)$-quasi basis are generalized Riesz systems.
- Application to non-self-adjoint Hamiltonians and relevant operators.

## Abstract

The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.