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Dilations of Gamma-contractions by solving operator equations

Bhattacharyya, Tirthankar and Pal, Sourav and Roy, Subrata Shyam (2011) Dilations of Gamma-contractions by solving operator equations. In: ADVANCES IN MATHEMATICS, 230 (2). pp. 577-606. PDF adv_math_230_577-606_2012.pdf - Published Version Restricted to Registered users only Download (258kB) | Request a copy
Official URL: http://dx.doi.org/10.1016/j.aim.2012.02.016

Abstract

For a contraction P and a bounded commutant S of P. we seek a solution X of the operator equation S - S*P = (1 - P* P)(1/2) X (1 - P* P)(1/2) where X is a bounded operator on (Ran) over bar (1 - P* P)(1/2) with numerical radius of X being not greater than 1. A pair of bounded operators (S, P) which has the domain Gamma = {(z(1) + z(2), z(2)): vertical bar z(1)vertical bar < 1, vertical bar z(2)vertical bar <= 1} subset of C-2 as a spectral set, is called a P-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Gamma-contraction (S, P). This allows us to construct an explicit Gamma-isometric dilation of a Gamma-contraction (S, P). We prove the other way too, i.e., for a commuting pair (S, P) with parallel to P parallel to <= 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S, P) is a Gamma-contraction. We show that for a pure F-contraction (S, P), there is a bounded operator C with numerical radius not greater than 1, such that S = C + C* P. Any Gamma-isometry can be written in this form where P now is an isometry commuting with C and C. Any Gamma-unitary is of this form as well with P and C being commuting unitaries. Examples of Gamma-contractions on reproducing kernel Hilbert spaces and their Gamma-isometric dilations are discussed. (C) 2012 Elsevier Inc. All rights reserved.

Item Type: Journal Article Copyright for this article belongs to Elsevier Science Gamma contractions;Spectral set;Gamma isometric dilation;Fundamental equation Division of Physical & Mathematical Sciences > Mathematics review EPrints Reviewer 22 Aug 2012 05:42 23 Aug 2012 04:36 http://eprints.iisc.ac.in/id/eprint/44430 View Item