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A generalized Weyl equidistribution theorem for operators, with applications


Authors: J. R. Blum and V. J. Mizel
Journal: Trans. Amer. Math. Soc. 165 (1972), 291-307
MSC: Primary 47A35; Secondary 28A65
DOI: https://doi.org/10.1090/S0002-9947-1972-0328633-9
MathSciNet review: 0328633
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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 Cesàro 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 Cesàro 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0328633-9
Keywords: Weak convergence of operator measures, ergodic theorem for spectral measures, convergence methods, generalized Riemann-Lebesgue Lemma
Article copyright: © Copyright 1972 American Mathematical Society

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