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Perturbations of Operators, Connections with Singular Integrals, Hyperbolicity and Entropy

In: Harmonic Analysis and Discrete Potential Theory

Author

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  • Dan Voiculescu

    (University of California, Department of Mathematics)

Abstract

We have studied in a series of papers ([10], [11], [12]) perturbations of Hilbert space operators using a certain invariant k J (τ), where J is a normed ideal of operators and τ is an n-tuple of operators. This number can be viewed as a “size J”- dimensional measure of τ. Frequently, evaluation of k J (τ) is related to the asymptotic of eigenvalues of certain singular integrals. In the case of translation operators in the regular representation of a discrete group G the number k J is related to the analogue of Yamasaki’s hyperbolicity condition on the Cayley graph of G with respect to the norm defining J. Quite recently, we have shown that in case J is the Macaev ideal $$C_\infty ^ - $$ , the invariant k J is related to the entropy of dynamical systems ([13]). Also in the case of the Macaev ideal, the existence of a random walk with positive entropy on a discrete group implies a hyperbolicity condition [14].

Suggested Citation

  • Dan Voiculescu, 1992. "Perturbations of Operators, Connections with Singular Integrals, Hyperbolicity and Entropy," Springer Books, in: Massimo A. Picardello (ed.), Harmonic Analysis and Discrete Potential Theory, pages 181-191, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4899-2323-3_14
    DOI: 10.1007/978-1-4899-2323-3_14
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