Operator-valued bases on Hilbert spaces

Subject Areas : Functional analysis

M. S. Asgari ¹ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 13185-768, Tehran, Iran)

Received: 2013-07-11 Accepted : 2013-12-25 Published : 2014-01-01

Keywords: ov-bases, dual ov-bases, Bessel ov-bases, Hilbert ov-bases, ov-biorthogonal sequence,

Abstract :

In this paper we develop a natural generalization of Schauder basis theory, we termoperator-valued basis or simply ov-basis theory, using operator-algebraic methods. We proveseveral 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 ofBessel, Hilbert ov-basis and obtain some characterizations of them. We study orthonormaland Riesz ov-bases for Hilbert spaces. Finally we consider the stability of ov-bases undersmall perturbations. We generalize a result of Paley-Wiener [4] to the situation of ov-basis.

References:

[1] M. S. Asgari, H. Rahimi, Generalized frames for operators in Hilbert spaces, Inf. Dim. Anal. Quant. Probab. Rel. Topics, Vol. 17, No. 2, (2014), 1450013-1 - 1450013-20.

[2] W. Rudin, Functional Analysis,McGrawHill. Inc, New York, (1991).

[3] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl. (2006), 322, 437-452.

[4] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, (2001).