Generalized Riesz Systems and Quasi Bases in Hilbert Space

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 { φn} and { ψn} , such that ∑n=0∞〈x,φn〉〈ψn,y〉=〈x,y〉 for all x∈ D and y∈ E. Second, it is shown that if biorthogonal sequences { φn} and { ψn} 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.