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].
Supported in part by National Science Foundation grant DMS 89-12362.
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Voiculescu, D. (1992). Perturbations of Operators, Connections with Singular Integrals, Hyperbolicity and Entropy. In: Picardello, M.A. (eds) Harmonic Analysis and Discrete Potential Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2323-3_14
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DOI: https://doi.org/10.1007/978-1-4899-2323-3_14
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