Linear Operator Theory

Instructor: Dr. J. A. Virtanen

Contact: majav@bristol.ac.uk

Schedule: October 11 - December 2011, Tuesdays 10:00 a.m. - 12:00 p.m.

Course notes: weeks 1-10

Exercises and Solutions: pdf (updated dec 12)

Grades: submitted! Solutions to the 10 problems are here. Thank you all for your participation.

Registration: email Margaret Sloper at graduate.studies@maths.ox.ac.uk

Assessment:

• concrete operators (e.g., Toeplitz, Hankel, singular integral, or Wiener-Hopf, composition operator) acting on a function space (e.g., Hardy, Bergman, or Fock space) and some of its (spectral) properties
• Riemann-Hilbert problems
• invariant subspace problem
• pseudospectrum
• Toeplitz and Hankel matrices
• multivariable operator theory
• Szegö's theorems
• C*-algebras
• (Hankel operators in) control theory
• unbounded operators on Hilbert space
• Schrödinger operators
• random matrix theory, orthogonal polynomials

Contents:

• preliminaries on functional analysis
• bounded linear operators on Hilbert spaces
• Banach algebras
• basic spectral theory
• compact operators
• Fredholm operators
• Schatten class operators
• examples of concrete linear operators:
  • singular integral operators
  • Toeplitz operators and matrices
  • composition operators
• applications of operator theory

Related literature:

• J. B. Conway, A course in functional analysis
• E. B. Davies, Linear operators and their spectra
• R. G. Douglas, Banach algebra techniques in operator theory
• N. Dunford and J. T. Schwartz, Linear operators
• D. E. Edmunds and W. D. Evans, Spectral theory and differential operators
• W. Rudin, Functional analysis
• K. Zhu, Operator theory in function spaces

Prerequisites:

intermediate analysis and linear algebra courses (functional analysis is useful but previous knowledge not necessary as preliminaries will be provided)

percy deift
boetther and jav
Hankel Operators and their Applications, V. Peller
Introduction to Large Truncated Toeplitz Matrices, A. Boettcher and B. Silbermann