Berger measures for the Schur product of weighted shifts

Abstract

In this paper we present a concrete Berger measure for the Schur product of weighted shifts. This measure will be seen as a convolution of their Berger measures. This is related to the p-th power problem for measures. For n ∈ N, the n-th power of a Berger measure is a convolution of all same measures and this case was solved partially by the author of this paper. We will extend this problem to a convolution of any Berger measures. We investigate a convolution of mutually distinct measures and then we discuss any combination of Berger measures, and any two more general weighted shifts. Since a Berger measure is closely related to subnormal weighted shifts, our result is helpful for the study of subnormality and add to the very small list of subnormal weighted shifts for which Berger measure is known concretely