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


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.


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.


Classical Articles

  1. [Hl]
    Hille, E.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., vol. 38. Am. Math. Soc., New York (1948) Google Scholar
  2. [St3]
    Stone, M.H.: On one-parameter unitary groups in Hilbert space. Ann. Math. 33, 643–648 (1932) CrossRefGoogle Scholar
  3. [Yo]
    Yosida, K.: On the differentiability and the representation of one-parameter semigroups of linear operators. J. Math. Soc. Jpn. 1, 15–21 (1949) CrossRefGoogle Scholar


  1. [D1]
    Davies, E.B.: One-Parameter Semigroups. Academic Press, London (1980) zbMATHGoogle Scholar
  2. [EN]
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, Berlin (2000) zbMATHGoogle Scholar
  3. [HP]
    Hille, E., Phillips, R.S.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., New York (1957) zbMATHGoogle Scholar
  4. [Hr]
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer-Verlag, Berlin (1973) Google Scholar
  5. [Kz]
    Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York (1968) zbMATHGoogle Scholar
  6. [Pa]
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin (1983) CrossRefzbMATHGoogle Scholar
  7. [RA]
    Rade, L., Westergren, B.: Springers Mathematische Formeln. Springer-Verlag, Berlin (1991) Google Scholar
  8. [RS4]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York (1978) zbMATHGoogle Scholar
  9. [Ru1]
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill Inc., New York (1974) zbMATHGoogle Scholar


  1. [Hs]
    Harish-Chandra: Representations of a semi-simple Lie group on a Banach space I. Trans. Am. Math. Soc. 75, 185–243 (1953) CrossRefzbMATHMathSciNetGoogle Scholar
  2. [Ka]
    Kaufman, W.F.: Representing a closed operator as a quotient of continuous operators. Proc. Am. Math. Soc. 72, 531–534 (1978) CrossRefzbMATHGoogle Scholar
  3. [Ne1]
    Nelson, E.: Analytic vectors. Ann. Math. 70, 572–614 (1959) CrossRefzbMATHGoogle Scholar
  4. [Nu]
    Nussbaum, A.E.: Quasi-analytic vectors. Ark. Math. 6, 179–191 (1965) CrossRefzbMATHMathSciNetGoogle Scholar
  5. [Ti]
    Tillmann, H.G.: Zur Eindeutigkeit der Lösungen der quantenmechanischen Vertauschungsrelationen. Acta Sci. Math. (Szeged) 24, 258–270 (1963) zbMATHMathSciNetGoogle Scholar
  6. [Tr]
    Trotter, H.: On the product of semigroups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959) CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Wo]
    Woronowicz, S.L.: Unbounded elements affiliated with C -algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

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

Personalised recommendations