B0323 - ANALISI FUNZIONALE E TEORIA DEGLI OPERATORI

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. Next, we will consider some spectral theorems for selfadjoint, and non-selfadjoint operators.

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.

Readings/Bibliography

Lecture notes on Virtuale.

Suggested readings:

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.ù

The exercise part is pass/fail. The grade is attributed according to the performance at the oral exam.

Teaching tools

The notes of the course and other material (articles, etcetera) will be made available on Virtuale. Here you can find last year's lecture notes: some lecture notes in analysis and probability

Office hours

See the website of Nicola Arcozzi