Random weighted shifts

摘要

In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights w(n) = 1 in the classical unilateral shift T, defined as Te-n = w(n)e(n)+1, where{e(n)}(n=1)(infinity) form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables {X-n}(n=1)(infinity) ; that is, w(n) = X-n. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet spaces in function-theoretic operator theory.