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 {phi n}{psi n}\\, such that n-ary sumation n=0 infinity x,phi n psi n,y=x,yfor all x is an element of Dy is an element of E. Second, 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.