Curvature invariants and Generalized Canonical operator models

Details Curvature invariants and Generalized Canonical operator models Curvature invariants and Generalized Canonical operator models

2010-09-16

arxiv.org/abs/1009.3258v1

Mathematics

Functional Analysis

21 pages

Scientific paper

There are two models for contraction operators on reproducing kernel Hilbert spaces, that of Sz.-Nagy-Foias introduced in the sixties and the later one from the late seventies due to M. Cowen and the first author. In comparing the two models, this paper interprets the former as a quotient Hilbert module of vector-valued Hardy spaces. Alongside this resolution is a resolution of hermitian anti-holomorphic vector bundles for which the curvatures can be calculated. Moreover, one can obtain other models replacing the Hardy space by other Hilbert spaces of holomorphic functions on the unit disk such as the weighted Bergman spaces. Further, one can decide when such quotient modules are unitarily equivalent and, perhaps, similar. In particular, it seems that the results are independent of the building block Hilbert spaces of holomorphic functions used. The techniques involved are a blend of complex geometry and harmonic analysis. In many cases, questions about the quotient Hilbert modules are reduced to questions involving anti-holomorphic sub-bundles of trivial finite-dimensional bundles over the disk.

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