Elements of spectral theory without the spectral theorem

David Krejčiřík*, Petr Siegl

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


This chapter is mainly devoted to a collection of basic facts from the spectral theory of operators in Hilbert spaces. It summarizes some efficient methods how to construct a closed operator with nonempty resolvent set. The chapter also talks about operators that are similar to self-adjoint (or more generally normal) operators. It recalls the notion of pseudospectra as more reliable information about non-self-adjoint operators than the spectrum itself and collects some abstract methods that can be effectively used to construct a quasi-m-accretive operator from a formal expression. Symmetric forms are familiar in quantum mechanics, where they have a physical interpretation of expectation values. For non-self-adjoint operators, a more general class of sectorial forms is needed. The theory of compact operators in Hilbert spaces is reminiscent of the theory of operators in finite-dimensional spaces. Highly non-self-adjoint operators have properties very different from self-adjoint or normal operators.

Original languageEnglish
Title of host publicationNon-Selfadjoint Operators in Quantum Physics
Subtitle of host publicationMathematical Aspects
Number of pages51
ISBN (Electronic)9781118855300
ISBN (Print)9781118855287
Publication statusPublished - 31 Jul 2015
Externally publishedYes


  • Closed operator
  • Hilbert spaces
  • Pseudospectra
  • Quantum mechanics
  • Quasi-m-accretive operator
  • Spectral theory

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Engineering(all)
  • Mathematics(all)


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