Advanced Topics in Complex Analysis
A.Y. 2023/2024
Learning objectives
Introduction to the more known spaces of holomorphic functions in the disc and in a half plane. Analysis of their properties, with attention to the proofs techniques. The focus is on Hardy and (weighted) Bergman spaces on the disc and in a half plane, Paley-Wiener spaces and Bernstein spaces.
Expected learning outcomes
Knowledge of the topics and results, and application to exercises that need also computational techniques.
Lesson period: Second semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
This course is devoted to the interplay with the spectral analysis of canonical systems and the associated spaces of entire functions. The tentative list of topics is the following:
-- Canonical systems: definitions, basic properties and examples.
-- Symmetric and self-adjoint operators and relations on a Hilbert space.
-- Spectral theory for operators on a Hilbert space, the spectral theorem.
-- Spectral representation of a canonical system.
-- Basic properties of entire functions.
-- Functions of exponential type, the Paley--Wiener space.
-- de Branges spaces.
-- Inverse spectral theory for a canonical system.
-- Canonical systems: definitions, basic properties and examples.
-- Symmetric and self-adjoint operators and relations on a Hilbert space.
-- Spectral theory for operators on a Hilbert space, the spectral theorem.
-- Spectral representation of a canonical system.
-- Basic properties of entire functions.
-- Functions of exponential type, the Paley--Wiener space.
-- de Branges spaces.
-- Inverse spectral theory for a canonical system.
Prerequisites for admission
Prerequisites are Analisi Complessa and Analisi Reale. Elementi di Analisi Funzionale is suggested.
Teaching methods
The course is offered as standard blackboard lectures.
Teaching Resources
-- L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall Inc.
-- C. Remling, Spectral Theory of Canonical Systems, De Gruyter Studies in Mathematics
-- R. Romanov, Canonical Systems and de Branges spaces, preprint arXiv:1408.6022v1
-- M. Peloso, Notes for the course.
-- C. Remling, Spectral Theory of Canonical Systems, De Gruyter Studies in Mathematics
-- R. Romanov, Canonical Systems and de Branges spaces, preprint arXiv:1408.6022v1
-- M. Peloso, Notes for the course.
Assessment methods and Criteria
The final examination consists of an oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Peloso Marco Maria
Educational website(s)
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica