## General information

Course type | AMUPIE |

Module title | Spectral Theory |

Language | English |

Module lecturer | Andrzej Sołtysiak |

Lecturer's email | asoltys@amu.edu.pl |

Lecturer position | profesor UAM |

Faculty | Faculty of Mathematics and Computer Science |

Semester | 2022/2023 (winter) |

Duration | 60 |

ECTS | 6 |

USOS code | 06-DTSPUM0-E |

## Timetable

## Module aim (aims)

The aim of the course is to present very important and classical result from operator theory such as the spectral theorem for self-adjoint operators on a Hilbert space.

## Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Background knowledge in mathematical analysis and functional analysis.

** **

## Syllabus

Week 1: Banach spaces and bounded linear operators on a Banach space– short reminder of main results.

Week 2: Hilbert spaces, the orthogonal projection theorem.

Week 3: Bounded linear operators on a Hilbert space, the adjoint of a bounded linear operator, normal, self-adjoint, and unitary operators.

Week 4: Invariant and reducing subspaces.

Week 5: The spectrum of a bounded linear operator on a Banach space.

Week 6: Main theorems concerning spectra.

Week 7: The spectrum of a bounded linear operator on a Hilbert space.

Week 8: The spectrum of a compact operator on a Banach space. The Riesz theorem.

Week 9: Integral operators. The Fredholm theorems.

Week 10: Orthogonal projection operators.

Week 11: The spectral theorem in a finite-dimensional space. The spectral theorem for normal compact operators.

Week 12: Positive operators.

Week 13: The functional calculus for self-adjoint operators.

Week 14: The spectral theorem for self-adjoint operators.

Week 15: The spectral theorem for normal and unitary operators.

## Reading list

- J. B. Conway,
*Course in Functional Analysis*, Springer-Verlag, New York 1997. - N. Dunford, J. T. Schwartz,
*Linear Operators, Part II*, Interscience Publishers, New York 1963. - P. R. Halmos,
*Introduction to Hilbert Space and the Theory of Spectral Multiplicity*, Chelsea Publ. Comp., New York 1951. - P. R. Halmos,
*A Hilbert Space Problem Book*, Springer-Verlag, New York 1982. - W. Mlak,
*Hilbert Spaces and Operator Theory*, PWN and Kluver, Warszawa and Dordrecht 1991. - F. Riesz, B. SZ.-Nagy,
*Functional Analysis*, Frederick Ungar Publ. Co., New York 1955. - W. Rudin,
*Functional Analysis*, McGraw-Hill Inc., New York 1991.