Generalized Riesz Systems and Quasi Bases in Hilbert Space

Abstract

The purpose of this article is twofold. First of all, the notion of $({\mathcal {D}}, {\mathcal {E}})$-quasi basis is introduced for a pair $({\mathcal {D}}, {\mathcal {E}})$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi _n \}$ and $\{ \psi _n \}$, such that $\sum _{n=0}^\infty \left\langle {x}, {\varphi _n}\right\rangle \left\langle {\psi _n}, {y}\right\rangle =\left\langle {x}, {y}\right\rangle$ for all $x \in {\mathcal {D}}$ and $y \in {\mathcal {E}}$. Second, it is shown that if biorthogonal sequences $\{ \varphi _n \}$ and $\{ \psi _n \}$ form a $({\mathcal {D}},{\mathcal {E}})$-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.