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Miscellanea

  • Konrad Schmüdgen
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)

Abstract

Chapter 7 is devoted to a number of important technical tools and special topics for the study of closed operators and self-adjoint operators. We begin with the polar decomposition of a densely defined closed operator. Then the polar decomposition is applied to the study of the operator relation A A=AA +I. Next, the bounded transform and its basic properties are developed. The usefulness of this transform has been already seen in the proofs of various versions of the spectral theorem in Chap.  5. Special classes of vectors (analytic vectors, quasi-analytic vectors, Stieltjes vectors) are studied in detail. They are used to derive criteria for the self-adjointness of symmetric operators (Nelson and Nussbaum theorems) and for the strong commutativity of self-adjoint operators. The last section of this chapter treats the tensor product of unbounded operators on Hilbert spaces.

Keywords

Quasi-analytic Vectors Strong Substitutability Polar Decomposition Important Technical Tool Symmetric Operator 
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 Dordrecht 2012

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  1. 1.Dept. of MathematicsUniversity of LeipzigLeipzigGermany

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