- Docente: Nicola Arcozzi
- Credits: 6
- SSD: MAT/05
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
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from Feb 20, 2024 to May 31, 2024
Learning outcomes
At the end of the course, the student in acquainted with some foundational objectcs and concepts and Operator Theory, including basic results from Spectral Theory, and with some of their applications within mathematics, or to science and technology.
Course contents
Basic problems of Spectral Theory are (i) the decomposition of a linear operator acting on a linear space into elementary operators acting on smaller subspaces, and the representation of the operator in some canonical form; (ii) the operator calculus: the construction of new operators from old ones. These operations are powerful tools in all branches of mathematics, and early in the XX century it appeared that they were foundational in the mathematical models of Quantum Mechanics. Moreover, they provide useful points of view on many mathematical landscapes.
We start with basic notions about operators on Banach spaces and their topologies, with examples, spectra of operators and their basic properties. The holomorphic calculus for bounded operators is presented. We then consider the Spectral Theorem for compact operators. Next, the Spectral Theorem for bounded, self-adjoint operators on a Hilbert space, and the more difficult Spectral Theorem for unbounded, self-adjoint operators are stated and proved.
Examples and applications of the theory will be presented all along the course. Their choice will also depend on the interests and inclinations of the students attending the class.
If there is time, we will see Beurling's Theorem on Invariant Subspaces as an example of what can happen and what can be done when the operator is not selfadjoint (not "normal", in fact).
Readings/Bibliography
Michael Reed, Barry Simon, Functional Analysis (1980)
Peter D. Lax, Functional Analysis (1966)
Jan van Neerven, Functional Analysis (2023) https://arxiv.org/pdf/2112.11166.pdf
Teaching methods
Blackboard lectures.
Assessment methods
During the course some homework exercises will be assigned and graded, mostly asking the student to work out the details of some examples. At the end of the course, there will be an oral exam.
Part of the contents to be examined might be varied for students having a special interests in a particular topic.
Teaching tools
The notes of the course and other material (articles, etcetera) will be made available on Virtuale.
Office hours
See the website of Nicola Arcozzi