Abstract
We arrive at the de Branges-Rovnyak space D(W) from the point of view of model theory, i. e., as the space associated with a canonical model for a general completely nonunitary contraction operator on a Hilbert space. Connections with systems theory make contact with the approach of Livsic and Brodskii, in particular, the role of the characteristic function as the transfer function of a unitary system. The connection between invariant subspaces and regular factorizations leads to the introduction of the overlapping spaces of de Branges-Rovnyak. This analysis applied to the particular factorization 0 = 0 · W · 0 leads to the derivation of the various geometric decompositions of the unitary dilation space for a contraction, the cornerstone of the model theory of Sz.-Nagy-Foias.
The first author was partially supported by National Science Foundation Grant DMS-8701625.
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Ball, J.A., Cohen, N. (1991). De Branges-Rovnyak Operator Models and Systems Theory: A Survey. In: Bart, H., Gohberg, I., Kaashoek, M.A. (eds) Topics in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5672-0_5
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DOI: https://doi.org/10.1007/978-3-0348-5672-0_5
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