On Some Operator Colligations and Associated Reproducing Kernel Pontryagin Spaces

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.