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The Structure of C1.-Contractions

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

Abstract

We systematically exploit the operators intertwining a given contraction with an isometry or unitary operator. Given operators T on \(\mathfrak{N}\) and T on \(\mathfrak{N^\prime}\), we denote by\(\mathfrak{N^\prime}\) g \((T, T^\prime)\)the set of all intertwining operators; these are the bounded linear transformations\(X: \mathfrak{N} \rightarrow \mathfrak{N^\prime}\) such that \(XT=T^\prime X\). We also use the notation {T} = ℐ(T,T) for the commutant of T. Fix a contraction T on \(\mathfrak{H}\) an isometry (resp., unitary operator) V on ℌ, and X∈ ℐ (T, V) such that ∥X∥ ≤ 1. The pair (X,V) is called an isometric (resp., unitary) asymptote of T if for every isometry (resp., unitary operator) V′, and every X′ ∈ ℐ(T, V′) with ∥X∥ ≤ 1, there exists a unique Y ∈ ℐ(V, V′) such that V′ = Y X and ∥Y′∥ ≤ 1.

Keywords

Invariant Subspace Separable Hilbert Space Cyclic Vector Spectral Mapping Theorem Unilateral Shift 
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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