Riesz basis for strongly continuous groups

TL;DR

This paper establishes conditions under which eigenvectors of a generator of a strongly continuous group form a Riesz basis in a Hilbert space, emphasizing the importance of eigenvalue gaps and spectral grouping.

Contribution

It proves that uniform eigenvalue gaps and dense eigenvector spans guarantee a Riesz basis, and explores spectral grouping conditions for basis formation.

Findings

Eigenvectors form a Riesz basis under uniform eigenvalue gaps.

Spectral grouping with bounded gaps leads to Riesz families.

Union of orthonormal bases in spectral ranges forms a Riesz basis.

Abstract

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space.