A Generalized Weyl Equidistribution Theorem for Operators, With Applications

Abstract

The present paper is motivated by the observation that Weyl’s equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl’s theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesà ro convergence property, equivalently, an “ergodic theorem", which is valid for all (projection-valued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with “strongly mixing” unitary transformations. The final sections are devoted to applications of the preceding results in the study of complex-valued Borel measures as well as to an extension of our results to summability methods other than Cesà ro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.