Harmonizability, V-Boundedness, and Stationary Dilation of Banach-Valued Processes

Abstract

The purpose of this work is to introduce concepts of stationary dilation, harmonizability, and V-boundedness for processes which take values in arbitrary Banach spaces and study their relationship. The paper extends for Banach spaces of cotype 2 the earlier work of ([N],[MS2]) and is different from ([MS1],[MI]) in that we are interested in obtaining a stationary dilation through a Hilbert space-valued stationary process. For this we use the spectral dilation of an L(x, y)-valued measure. A general result concerning such a spectral dilation which includes the results of ([R],[MS1],[MI]) is given in [MR]. In this paper we present an elementary proof of the equivalence theorem of [R] which forms the basis for the main results of ([MS1],[MI]). The stationary dilation so obtained is then applied to SαS processes. In addition, some observations on general Banach space-valued processes are given. We begin with the basic definitions, results, and notation.