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Spectral Decompositions of Self-adjoint and Normal Operators

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

Abstract

Chapter 5 is devoted to the spectral decomposition of self-adjoint and normal operators. In the first section, the spectral theorem for a single bounded self-adjoint operator is proved. Then the spectral theorem for an unbounded self-adjoint operator is derived from the bounded case by using the bounded transform T(I+T T)−1/2. The spectral integrals with respect to the corresponding spectral measure are considered as functions of the self-adjoint operator. This functional calculus and a number of important applications (spectrum, fractional powers, Stone’s formulas) are developed. Self-adjoint operators with simple spectra are studied. For an n-tuple of strongly commuting unbounded normal operators, the spectral theorem is proved, and the joint spectrum is defined and investigated. Permutability problems involving unbounded self-adjoint or normal operators are considered, and a number of equivalent characterizations of the strong commutavitity in terms of spectral measures, resolvents, and bounded transforms are given.

Keywords

Strongly Commuting Unbounded Normal Operators Spectral Theorem Bounded Self-adjoint Operator Unbounded Norm 
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.

References

Classical Articles

  1. [HT]
    Hellinger, E., Toeplitz, O.: Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten. Enzyklopädie d. Math. Wiss. II.C 13, 1335–1616 (1928) Google Scholar
  2. [Hi]
    Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner-Verlag, Leipzig (1912) zbMATHGoogle Scholar
  3. [Ri1]
    Riesz, F.: Les systèmes d’équations linéaires à une infinité d’inconnues. Gauthiers–Villars, Paris (1913) zbMATHGoogle Scholar
  4. [Ri2]
    Riesz, F.: Über die linearen Transformationen des komplexen Hilbertschen Raumes. Acta Sci. Math. Szeged 5, 23–54 (1930) zbMATHGoogle Scholar
  5. [St2]
    Stone, M.H.: Linear Transformations in Hilbert Space. Am. Math. Soc., New York (1932) Google Scholar
  6. [vN1]
    Von Neumann, J.: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102, 49–131 (1929) CrossRefzbMATHGoogle Scholar

Books

  1. [AG]
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Ungar, New York (1961) zbMATHGoogle Scholar
  2. [Bn]
    Berberian, S.K.: Notes on Spectral Theory. van Nostrand, Princeton (1966) zbMATHGoogle Scholar
  3. [BSU]
    Berezansky, Y.M., Sheftel, Z.G., Us, G.F.: Functional Analysis, vol. II. Birkhäuser-Verlag, Basel (1996) CrossRefzbMATHGoogle Scholar
  4. [BS]
    Birman, M.S., Solomyak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Kluwer, Dordrecht (1987) Google Scholar
  5. [Cw]
    Conway, J.B.: A Course in Functional Analysis. Springer-Verlag, New York (1990) zbMATHGoogle Scholar
  6. [DS]
    Dunford, N., Schwartz, J.T.: Linear Operators, Part II. Spectral Theory. Interscience Publ., New York (1963) zbMATHGoogle Scholar
  7. [RS1]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York (1972) Google Scholar
  8. [RN]
    Riesz, F., Nagy, Sz.-B.: Functional Analysis. Dover, New York (1990) zbMATHGoogle Scholar
  9. [Ru3]
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973) zbMATHGoogle Scholar

Articles

  1. [Cd1]
    Coddington, E.A.: Formally normal operators having no normal extensions. Can. J. Math. 17, 1030–1040 (1965) CrossRefzbMATHMathSciNetGoogle Scholar
  2. [Mc]
    McCarthy, C.A.: c p. Isr. J. Math. 5, 249–271 (1967) CrossRefzbMATHGoogle Scholar
  3. [Ne1]
    Nelson, E.: Analytic vectors. Ann. Math. 70, 572–614 (1959) CrossRefzbMATHGoogle Scholar
  4. [Sch3]
    Schmüdgen, K.: On commuting unbounded selfadjoint operators. Acta Sci. Math. (Szeged) 47, 131–146 (1984) zbMATHMathSciNetGoogle Scholar
  5. [Sn]
    Steen, L.A.: Highlights in the history of spectral theory. Am. Math. Monthly 80, 359–381 (1973) CrossRefzbMATHMathSciNetGoogle Scholar

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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