Random weighted shifts

Abstract

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 $Te_n=w_ne_{n+1}$, where $\{e_n\}_{n=1}^\infty$ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables $\{X_n\}_{n=1}^{\infty}$; 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. We calculate the spectra and determine their fine structures (Section 3). We classify the samples up to four equivalence relationships (Section 4). We introduce a family of random Hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces (Section 5). We compare them with three types of classical operators (Section 6); this is achieved in the form of generalized von Neumann inequalities. The invariant subspaces are shown to admit arbitrarily large indices and their semi-invariant subspaces model arbitrary contractions almost surely. We discuss a Beurling-type theorem (Section 7). We determine various non-selfadjoint algebras generated by $T$ (Section 8). Their dynamical properties are clarified (Section 9). Their iterated Aluthge transforms are shown to converge (Section 10). In summary, they provide a new random model from the viewpoint of probability theory, and they provide a new class of analytic functional Hilbert spaces from the viewpoint of operator theory. The technical novelty in this paper is that the methodology used draws from three (largely separate) sources: probability theory, functional Hilbert spaces, and the approximation theory of bounded operators.