Operator-valued bases on Hilbert spaces

Abstract

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also de ne the concepts of Bessel, Hilbert ov-basis and obtain some characterizations of them. We study orthonormal and Riesz ov-bases for Hilbert spaces. Finally we consider the stability of ov-bases under small perturbations. We generalize a result of Paley-Wiener [4] to the situation of ov-basis.