Abstract
The spaces ℌ(S), ℌ(\( \widetilde{S} \) ), and D(S) defined in Chapter 2 have many special properties when S(z) belongs to S K (F,B). Invariance under the difference-quotient transformation and an inequality characterize spaces of the form ℌ(S) for such functions (§3.1). Various conditions are derived for the canonical coisometric and isometric colligations to be unitary (§3.2, §3.3). Natural mappings relate the three spaces ℌ(S), ℌ(\( \widetilde{S} \) ), and D(S) and lead to an operator range characterization of D(S) in §3.4. A number of examples and applications are discussed in §3.5, including rational functions with unitary values on the unit circle.
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© 1997 Springer Basel AG
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Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H. (1997). The State Spaces. In: Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces. Operator Theory Advances and Applications, vol 96. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8908-7_3
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DOI: https://doi.org/10.1007/978-3-0348-8908-7_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9823-2
Online ISBN: 978-3-0348-8908-7
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