Proceedings of the London Mathematical Society Advance Access originally published online on March 25, 2009
Proceedings of the London Mathematical Society 2009 99(3):557-584; doi:10.1112/plms/pdp011
| ||||||||||||||||||||||||||||||||||||||||||||||||
© 2009 London Mathematical Society
The curvature invariant for a class of homogeneous operators
Indian Institute of Science
Bangalore 560 012
India
Indian Statistical Institute
R. V. College Post
Bangalore 560 059
India
ssroy@isibang.ac.in
Received 22 June 2007.
For an operator T in the class Bn(
), 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 ET 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 
Bn(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in Bn(
). 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.
2000 Mathematics Subject Classification 47B32 (primary), 46E22, 22D10, 22F50 (secondary).
The research of the first author was supported in part by a grant from the DST–NSF Science and Technology Cooperation Programme. The second author was supported by the Indian Statistical Institute.