Perturbations Of Bessel Sequences Of Order P In A Banach Space

Abstract:In this article, we first introduce the concepts of the Bessel sequences of order p in a Banach space, discuss their perturbations and establish the new perturbation theory, give two general forms of the 2×2 unitary matrices and confirm their equivalence, and discuss the structure question of the wavelet operator pair, which give a new way for constructing orthogonal wavelets in H (?) H.This article is divided into three chapters, the details are as follows:In Chapter 1, we first introduce the history of the theory on frame in a Banach Space, At the same time, we outline the recent development of wavelet analysis and the significance of the theory in other subjects. Then we give the concepts and basic properties of Bessel sequences of order p in a Banach space. Finally, we define the wavelet operator pairs and orthogonal wavelets and list the prime task of this article briefly.In Chapter 2, for a Bessel sequence f={fi}i∈I of order p(1< p<∞) in a Banach space X, we define a bounded linear operator Tf:X*→lp and establish a linear isometry isomorphism a from the space BXP(Ⅰ) of all Bessel sequences of order p in a Banach space X into the operator space B(X*,lp) so thatα(f)= Tf. In light of operator theory, we discuss perturbations of Bessel sequences of order p. In the second section, Operator Theory is used to discuss perturbations of Bessel sequences of order p(1≤p≤∞) in a Banach space, some sufficient conditions for a Bessel sequence of order p to be a Bessel sequence of order q under a perturbation are obtained and some sufficient conditions for a Bessel sequence of order 1 to be a Bessel sequence of order 1 and∞under a perturbation are given, respectively, and then some results are obtained.In Chapter 3, we give two general forms of 2×2 unitary matrices and discuss some necessary and sufficient conditions on two unitary matrices T and D satisfying the equality TD= DT2. Finally, we construct unitary operators D and T on H (?) H in order to obtain wavelet operator pairs on H (?) H, which give a new way for constructing orthogonal wavelets in H (?) H.