Generalized Riesz systems and quasi bases in Hilbert space

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 $\{{\phi }_n\}$ and $\{{\psi }_n\}$ 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 $\{{\phi }_n\}$ and $\{{\psi }_n\}$ 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.

submitted to MJOM