Dottorato di Ricerca in Matematica

Operator and Spectral Theory

a.a. 2017/18

Aggiornato il 3 novembre 2017.



Anno accademico: 2017/18

Docente: Marco Peloso
Ore di didattica: 25 ore di lezione (5CFU)
Periodo delle lezioni: 8 gennaio - 28 febbraio 2018
Calendario delle lezioni: in Aula Dottorato (Dip. di Matematica)

  • mercoledì 10/1, 10.30 - 12.30, giovedì 11/1, 14.30 - 16.30;
  • mercoledì 17/1, 10.30 - 12.30;
  • mercoledì 24/1, 10.30 - 12.30, giovedì 25/1, 14.30 - 16.30;
  • mercoledì 31/1, 10.30 - 12.30, giovedì 1/2, 14.30 - 16.30;
  • mercoledì 7/2, 10.30 - 12.30, giovedì 8/2, 14.30 - 16.30;
  • mercoledì 14/2, 10.30 - 12.30;
  • mercoledì 21/2, 10.30 - 12.30, giovedì 22/2, 14.30 - 16.30.

 


 

Syllabus

  • Basic facts of functional analysis:
    • Function spaces
    • Topological vector spaces
    • Banach and Hilbert spaces
  • Three fundamental theorems:
    • The Banach--Steinhaus Theorem
    • The open mapping theorem
    • Closed graph theorem
  • Bounded operators:
    • Topologies on bounded operators
    • Adjoints
    • Positive and unitary operators, and the polar decomposition
    • The spectrum
  • Compact operators
    • Trace class and Hilbert--Schimdt operators
    • The Schatten classes
  • The spectral theorem
    • Projection-valued measures
    • The spectral measure
  • Unbounded operators
    • Domains, adjoints, spectrum
    • Symmetric and self-adjoint operators
    • The spectral theorem
    • Examples and counter-examples
    • The Laplacian
    • Nelson's example
  • Some research problems
    • Buondedness of spectral multipliers for elliptic and subelliptic operators
    • On the existence of invariant subspces
Bibliography
  • W. Rudin, Functional Analysis, 3ed, McGraw Hill, 1995.
  • K. Yoisida, Functional Analysis, 6ed, Springer-Verlag, 1991.
  • M. Reed and B. Simon, Methods of Moder Mathematical Physics, 1 Functional Analysis, Academic Press 1972.
  • Notes of the course, part 1, part 2.

 


 

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