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Some Classes of Unbounded Operators

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

Abstract

In Chap. 3, we introduce and begin the study of some fundamental classes of unbounded operators. The most important ones for this book are symmetric operators and self-adjoint operators. For a densely defined symmetric operator, the deficiency indices are defined and investigated, and the von Neumann formula about the domain of the adjoint is obtained. As a consequence, some self-adjointness criteria for symmetric operators are derived. Among others, we give a short proof of Naimark’s classical theorem, which states that each densely defined symmetric operator has a self-adjoint extension on a possibly larger Hilbert space. Further, we define classes of operators (sectorial operators, accretive operators, dissipative operators) that will appear later as operators associated with sectorial forms or as generators of contraction semigroups. The last section of this chapter contains a brief introduction to unbounded normal operators.

Keywords

Unbounded Normal Operators Symmetric Operator Self-adjointness Criterion Accretive Operators Deficiency Indices 
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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