Generalized Riesz systems and quasi bases in Hilbert space

The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasibasisisintroducedforapair$(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{{\phi }_n\}$ and ${ \psi_n }$suchthat$\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$forall$x \in D$and$y \in E$. Secondly, it is shown that if biorthogonal sequences $\{{\phi }_n\}$ and ${ \psi_n }$forma$(D ,E)$-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.