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

Autor: Hans Zwart

Fuente: https://arxiv.org/

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