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Curvature Inequalities for Operators in the Cowen-Douglas Class of a Planar Domain

Reza, Md Ramiz (2018) Curvature Inequalities for Operators in the Cowen-Douglas Class of a Planar Domain. In: INDIANA UNIVERSITY MATHEMATICS JOURNAL, 67 (3). pp. 1255-1279.

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Official URL: https://dx.doi.org/10.1512/iumj.2018.67.7320


Fix a bounded planar domain Omega. If an operator T, in the Cowen-Douglas class B-1(Omega), admits the compact set Omega as a spectral set, then the curvature inequality K-T(w)<=-4 pi S-2(Omega)(w,w)(2), where S-Omega is the Szego kernel of the domain Omega, is evident. Except when Omega is simply connected, the existence of an operator for which K-T(w) = -4 pi S-2(Omega)(w,w)(2) for all w in Omega is not known. However, if w is a fixed but arbitrary point in Omega, then there exists a bundle shift of rank one, say S, depending on this w, such that K-s*(w) = -4 pi S-2(Omega)(w,w)(2). We prove that these extremal operators are uniquely determined. If T-1 and T-2 are two operators in B-1(Omega), each of which is the adjoint of a rank-one bundle shift, and K-T1(w) = -4 pi S-2(Omega)(w,w)(2) = KT2(w) for a fixed w in Omega, then T-1 and T-2 are unitarily equivalent. A surprising consequence is that the adjoints of only some of the bundle shifts of rank one occur as extremal operators in domains of connectivity greater than 1. These are described explicitly.

Item Type: Journal Article
Additional Information: Copyright of this article belong to INDIANA UNIV MATH JOURNAL, SWAIN HALL EAST 222, BLOOMINGTON, IN 47405 USA
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 20 Jul 2018 16:37
Last Modified: 20 Jul 2018 16:37
URI: http://eprints.iisc.ac.in/id/eprint/60261

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