Riesz Bases in Krein Spaces

TL;DR

This paper investigates Riesz bases in Krein spaces, establishing their properties, biorthogonal sequences, Gram matrices, and conditions for boundedness and invertibility, thus extending classical basis theory to indefinite inner product spaces.

Contribution

It introduces and characterizes Riesz bases in Krein spaces, including biorthogonal sequences and Gram matrix conditions, providing new insights into their structure and properties.

Findings

Defined Riesz bases in Krein spaces and their properties.

Characterized biorthogonal sequences and Gram matrices.

Established conditions for boundedness and invertibility of Gram matrices.

Abstract

We start by introducing and studying the definition of a Riesz basis in a Krein space $(\mathcal{K},[.,.])$, along with a condition under which a Riesz basis becomes a Bessel sequence. The concept of biorthogonal sequence in Krein spaces is also introduced, providing an equivalent characterization of a Riesz basis. Additionally, we explore the concept of the Gram matrix, defined as the sum of a positive and a negative Gram matrices, and specify conditions under which the Gram matrix becomes bounded in Krein spaces. Further, we characterize the conditions under which the Gram matrices $\{[f_n,f_j]_{n,j \in I_+}\}$ and $\{[f_n,f_j]_{n,j \in I_-}\}$ become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices.