Spectral Decompositions of Self-adjoint and Normal Operators

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


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.


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