Random Weighted Shifts on Hilbert Spaces of Analytic Functions

Let G be a bounded region in the complex plane and H be a Hilbert space of analytic functions on G satisfying (a) zf. H for all f is an element of H, and (b) {zn}(infinity)(n=0) forms an orthogonal basis of H with limn parallel to z(n+1)parallel to/parallel to z(n)parallel to = 1. This paper aims to study a random counterpart of the shift operator on H. We replace the weights w(n) = 1 in the shift operator on H, defined as Tz(n) = w(n)z(n+1), by a sequence of i.i.d. random variables {X-n}(infinity)(n=0); that is, w(n) = X-n. We call T a random weighted shift on H with weights {Xn}(infinity)(n=0). In this paper we classify the samples of T according to five usual equivalence relations. We compare random weighted shifts acting on various Hilbert spaces of analytic functions, via a binary relation (sic) borrowed from the representation theory of C*-algebras. Also we discuss the random Hardy space associated with T and, in certain case, determine the spectral picture of its multipliers.