Journal of Operator Theory

Volume 60, Issue 1, Summer 2008 pp. 3-28.

Nagy-Foias type functional models of nondissipative operators in parabolic domains

Authors: Dmitry V. Yakubovich
Author institution: Department of Mathematics, Universidad Autonoma de Madrid, Spain

Summary: A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space $X$ is constructed. This model is analogous to the Nagy-Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from $X$ to the model space and the analogue of the characteristic function are given explicitly. All usual consequences of the Nagy-Foias construction (the $H^\infty$ calculus, the commutant lifting, etc.) hold true in our context.

Keywords: A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space $X$ is constructed. This model is analogous to the Nagy-Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from $X$ to the model space and the analogue of the characteristic function are given explicitly. All usual consequences of the Nagy-Foias construction (the $H^\infty$ calculus, the commutant lifting, etc.) hold true in our context.

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