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arXiv · 1907.05604

Generalized Riesz systems and quasi bases in Hilbert space

Fabio Bagarello·Hiroshi Inoue·Camillo Trapani
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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 $\{ φ_n \}$ and $\{ ψ_n \}$ such that $\sum_{n=0}^\infty \ip{x}{φ_n}\ip{ψ_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ φ_n \}$ and $\{ ψ_n \}$ 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.

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Fabio Bagarello, Hiroshi Inoue, Camillo Trapani. 2019-07-12. Generalized Riesz systems and quasi bases in Hilbert space. https://arxiv.org/abs/1907.05604

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