The curvature invariant for a class of homogeneous operators

Misra, Gadadhar and Roy, Subrata Shyam (2009) The curvature invariant for a class of homogeneous operators. In: Proceedings London Mathematical Society, 99 (Part 3). pp. 557-584.

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Abstract

For an operator T in the class B-n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E-T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T is an element of B-n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B-n(). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Item Type:	Journal Article
Publication:	Proceedings London Mathematical Society
Publisher:	London Mathematical Society
Additional Information:	Copyright for this article belongs to London Mathematical Society.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	08 Dec 2009 08:42
Last Modified:	19 Sep 2010 05:52
URI:	http://eprints.iisc.ac.in/id/eprint/24944

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