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Models

In: An Introduction to Models and Decompositions in Operator Theory

Author

Listed:
  • Carlos S. Kubrusly

    (Catholic University — PUC/RJ
    National Laboratory for Scientific Computation — LNCC)

Abstract

An operator is said to be a universal model if every operator in B[H] is similar to a multiple of a part of it. Drop “universal” by constraining the definition to a class of operators: an operator is a model for a class of operators in B[H] if every operator in that class is similar to a multiple of a part of it. Equivalently (see Proposition 1.1), if every operator in that class is a part of an operator similar to a multiple of it. An operator is an extension of an operator T ∈ B[H] if T is a part of it. We shall say that a class P of operators is a pattern for a class C in B[H] if every operator in C has an extension in P (i.e. if every operator in C is a part of an operator in P). For instance, normal operators are patterns for subnormal operators (by the very definition of subnormal operators), bilateral shifts are patterns for unilateral shifts (cf. Lemma 2.14), and unitaries are patterns for isometries (according to Lemma 5.7). Thus an operator is a model for a class of operators C in B[H] if the class of all operators similar to a multiple of it is a pattern for C. Every uniformly stable operator is similar to a part of a canonical backward unilateral shift.

Suggested Citation

  • Carlos S. Kubrusly, 1997. "Models," Springer Books, in: An Introduction to Models and Decompositions in Operator Theory, chapter 0, pages 87-100, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-1998-9_7
    DOI: 10.1007/978-1-4612-1998-9_7
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