## ANNOUNCEMENTS

- Registration for Fall term opens Thursday, May 19, 2016.
- This page was last updated: .

## COURSE DESCRIPTION

The spectral theory of linear operators plays an important role in Analysis. Using the theory of Banach algebras we will prove the spectral theorem that provides conditions under which a bounded operator can be diagonalized. We will subsequently generalize the results to unbounded operators and discuss some applications. In the last part of the course the theory of semi-groups of operators will be introduced.

Among the topics covered in this course are: Banach algebras, basic properties of spectra, the spectral theorem, unbounded operators, the Cayley transform, semi-groups of operators, the Hille—Yosida theorem.

## POLICIES

### Grades

pass/fail

### Homework Policy

The examination will take place in the form of student talks on related subjects.

## TOPICS COVERED

- Banach algebras
- Basic properties of spectra
- Gelfand—Naimark theorem
- Spectral theorem for bounded normal operators
- Unbounded operators
- Cayley transform
- Spectral theorem for unbounded self-adjoint operators
- Semi-groups of operators
- Hille—Yosida theorem

## TEXTBOOKS

[1] Rudin, W.: Functional Analysis, ISBN-13: 978-0070542365

Additional resources:

[2] Simon, B.: Operator Theory: A Comprehensive Course in Analysis, Part 4, ISBN-13: 978-1470411039

[3] Akhiezer, N. I. and Glazman, M.: Theory of Linear Operators in Hilbert Space, ISBN-13: 978-0486677484

## LECTURE NOTES

Date | Description | Reference |
---|---|---|

09/27/16 | Banach spaces, Bounded operators, Banach Algebras | [1] pp. 245-252 |

09/29/16 | Complex Homomorphisms | [1] pp. 249-253 |

10/04/16 | Basic properties of spectra, Gelafnd--Mazur theorem, Symbolic Calculus | [1] pp. 253-262 |

10/06/16 | Spectral mapping theorem, Roots and logarithm, Ideals and homomorphisms of commutative Banach Algebras |
[1] pp. 262-277 |

10/11/16 | Gelfand transforms, B* Algebras | [1] pp. 277-286 |

10/13/16 | Gelfand--Naimark Theorem | [1] pp. 287-290 |

10/18/16 | Hermitian square roots, Applications to non-commutative algebras | [1] pp. 290-296 |

10/20/16 | Resolutions of the identity, Spectral measure | [1] pp. 316-321 |

10/25/16 | The spectral theroem for bounded normal operators, Theorem of Fuglede--Putnam--Rosenblum |
[1] pp. 325-321 [1] pp. 315-316 |

10/27/16 | Applications of the spetral theorem for bounded operators: Characterisation of Fourier--Stiltjes coefficients, Ergodic systems, Bohr's theorem for almost periodic functions |
[2] pp. 633-634 [1] pp. 339-340 [3] pp. 132-138 |

11/01/16 | Bohr's theorem for almost periodic functions, The spectrum of normal operators |
[3] pp. 132-138 [1] pp. 326-329 |

11/03/16 | Spectrum of compact operators, Square roots of positive
operators, Measure/multiplication operator version of the spectral theorem |
[1] pp. 329-333 [2] pp. 293-295 |

11/08/16 | Measure/multiplication operator version of the spectral
theorem Unbounded operators |
[2] pp. 294-295 [1] pp. 347-349 |

11/10/16 | Momentum operator, Symmetric operators, Self-adjoint operators | [1] pp. 349-354 |

11/15/16 | Maximally symmetric operators, Cayley transform | [1] pp. 354-358 |

11/17/16 | Cayley transform, Deficiency Indices, Self-adjoint extensions | [1] pp. 358-361 |

11/22/16 | Resolution of the identity, Funcitonal calculus for measurable
functions, Resolvent set, Point/Continuous/Residual sepctrum |
[1] pp. 361-367 |

11/24/16 | Thanksgiving | |

11/29/16 | Spectral Theorem for (unbounded) self-adjopint operators Dirichlet Laplacian, Neumann Laplacian, Berezin--Li--Yau inequality, Semi-groups |
[1] pp. 368-369 A. Laptev, J. Funct. Anal. (1997) [1] p. 375 |

12/01/16 | Hille--Yosida theorem | [1] pp. 376-382 |

12/06/16 | Student talks | |

12/08/16 | End of term |