On Some Operator Colligations and Associated Reproducing Kernel Pontryagin Spaces, Journal of Functional Analysis

Document Type

Article

Publication Date

1996

Abstract

Letsbe a Schur function, that is a function analytic and contractive in the unit disk D. Then the function 1−s(z) s(ω)*/1−zω* is positive in D. L. de Branges and J. Rovnyak proved that the associated reproducing kernel Hilbert space provides the state space for a coisometric realization ofs. In a previous work we extended this result to the case of operator valued functions with the denominator 1−zω* replaced bya(z) a(ω)*−b(z) b(ω)*, whereaandbare analytic functions subject to some conditions. In the present work we remove the positivity condition and allow the kernel to have a number of negative squares. Moreover, we consider functions whose values are bounded operators between Pontryagin spaces with the same index. We show that there exist reproducing kernel Pontryagin spaces which provide unitary, isometric, and coisometric realizations of the function. We also study the projective version of the above kernel.

Comments

This article was originally published in Journal of Functional Analysis, volume 136, in 1996. DOI: 10.1006/jfan.1996.0021

Peer Reviewed

1

Copyright

Elsevier

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