## Abstract

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 language | English |
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Title of host publication | Non-Selfadjoint Operators in Quantum Physics |

Subtitle of host publication | Mathematical Aspects |

Publisher | Wiley |

Chapter | 5 |

Pages | 241-291 |

Number of pages | 51 |

ISBN (Electronic) | 9781118855300 |

ISBN (Print) | 9781118855287 |

DOIs | |

Publication status | Published - 31 Jul 2015 |

Externally published | Yes |

## Keywords

- 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)