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Operator-Valued Analytic Functions

  • Béla Sz.-Nagy
  • Hari Bercovici
  • Ciprian Foias
  • László Kérchy
Chapter
Part of the Universitext book series (UTX)

Abstract

For any separable Hilbert space \(\mathfrak{U}\) we denote by L2(\(\mathfrak{U}\)) the class of functions \(v(t) (0 \leq t \leq 2\pi)\) with values in \(\mathfrak{U}\), measurable1 (strongly or weakly, which are equivalent due to the separability of \(\mathfrak{U}\)) and such that.
$$\parallel v \parallel ^{2} = \frac{1}{2\pi} \int \limits ^{2\pi}_{0} \parallel v(t) \parallel^{2}_{\mathfrak{U}} dt < \infty.V$$
(1.1)
With this definition of the norm \(\parallel v \parallel, L^{2}({\mathfrak{U}})\) becomes a (separable) Hilbert space; it is understood that two functions in \(L^{2}({\mathfrak{U}})\) are considered identical if they coincide almost everywhere (with respect to Lebesgue measure). If dim \({\mathfrak{U}} = ( {\rm i.e.,} {\rm if} L^{2}({\mathfrak{U}})\) consists of scalar-valued functions), we write L 2 instead of \(L^{2}({\mathfrak{U}}).\)

Keywords

Hilbert Space Analytic Function Scalar Multiple Invariant Subspace Linear Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Béla Sz.-Nagy
  • Hari Bercovici
    • 1
  • Ciprian Foias
    • 2
  • László Kérchy
    • 3
  1. 1.Mathematics DepartmentIndiana UniversityBloomingtonUSA
  2. 2.Mathematics DepartmentTexas A & M UniversityCollege StationUSA
  3. 3.Bolyai InstituteSzeged UniversitySzegedHungary

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