New Characterizations of Operator-Valued Bases on Hilbert Spaces

Abstract

In this paper we study operator valued bases on Hilbert spaces and similar to Schauder bases theory we introduce characterizations of this generalized bases in Hilbert spaces. We redefine the dual basis associated with a generalized basis and prove that the operators of a dual $g$-basis are continuous. Finally we consider the stability of $g$-bases under small perturbations. We generalize two results of Krein-Milman-Rutman and Paley-Wiener [7] to the situation of $g$-basis.