Non-self-adjoint hamiltonians defined by generalized Riesz bases

TL;DR

This paper explores operators defined by generalized Riesz bases, extending previous work on Riesz bases and their connection to non-self-adjoint Hamiltonians in quasi-Hermitian quantum mechanics.

Contribution

It introduces the concept of generalized Riesz bases and studies operators defined by them, broadening the framework beyond traditional Riesz bases.

Findings

Almost all results from previous Riesz basis work become trivial under the new framework.

Generalized Riesz bases extend the class of operators related to non-self-adjoint Hamiltonians.

Provides new insights into the structure of operators in quasi-Hermitian quantum mechanics.

Abstract

In \cite{b-i-t}, F. Bagarello, A. Inoue and C. Trapani investigated some operators defined by Riesz bases. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. In this paper, we change the frameworks of these operators, and then almost results obtained in \cite{b-i-t} become trivial. Furthermore, we introduce a notion of generalized Riesz bases which is a generalization of Riesz bases and investigate some operators defined by generalized Riesz bases.