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

Manifold 

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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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