Operator-Valued Analytic Functions

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


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.
$$\parallel v \parallel ^{2} = \frac{1}{2\pi} \int \limits ^{2\pi}_{0} \parallel v(t) \parallel^{2}_{\mathfrak{U}} dt < \infty.V$$
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 \(L^{2}({\mathfrak{U}}).\)


Hilbert Space Analytic Function Scalar Multiple Invariant Subspace Linear Manifold 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Béla Sz.-Nagy
  • Hari Bercovici
    • 1
    Email author
  • 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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