Aggiornato
il 3 novembre 2017.
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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.
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Syllabus
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- 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
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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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home
marco.peloso@unimi.it
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