Harnack and Shmul’yan pre-order relations for Hilbert space contractions |
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Institution: | 1. Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59655 Villeneuve d’Ascq Cedex, France;2. Department of Mathematics, “Lucian Blaga” University of Sibiu, Dr. Ion Ra?iu 5-7, Sibiu, 550012, Romania |
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Abstract: | We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul’yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul’yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul’yan parts coincide. We also discuss an extension, recently considered by S. ter Horst (2014), of the Shmul’yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul’yan–ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined. |
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Keywords: | Harnack pre-order Shmul’yan pre-order Hilbert space contractions Asymptotic limit Quasi-normal operators Partial isometries Toeplitz operators |
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