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Description

This couse provides an introduction to analysis in infinite dimensions with a minimum of prerequisites. The core of the course concerns operators on a Hilbert space including the continuous functional calculus for bounded selfadjoint operators. There will be an emphasis on positivity and on matrices of operators.
The course includes some basic introductory material on Banach spaces and Banach algebras. It also includes some elementary (infinite dimensional) linear algebra that is usually excluded from undergraduate curricula.
Here is a very brief list of the many further topics that this course looks forward to.
Banach space theory and Banach algebras; C*-algebras, von Neumann algebras and operator spaces (which may be viewed respectively as noncommutative topology, noncommutative measure theory and `quantised' functional analysis); Hilbert C*-modules; noncommutative probability (e.g. free probability), the theory of quantum computing, dilation theory;Unbounded Hilbert space operators, one-parameter semigroups and Schrodinger operators. And that is without starting to mention Applied Maths and Statistics applications ...
Relevant books
  • G. K. Pederson, Analysis Now (Springer, 1988)
    [This course may be viewed as a preparation for studying this text (which is already a classic).]
  • Simmonds, Introduction to Topology and Modern Analysis (McGraw-Hill, 1963)
    [Covers far more than the course, but is still distinguished by its great accessibility.]
  • P.R. Halmos, Hilbert Space Problem Book (Springer, 1982)
    [Collected and developed by a master expositor.]
There are many many other books which cover the core part of this course.

Semester

Autumn 2009 (Monday, October 5 to Friday, December 11)

Timetable

  • Tue 14:05 - 14:55
  • Wed 14:05 - 14:55

Prerequisites

Standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology.

Syllabus

I Preliminaries (5 lectures)
  1. Linear algebra, including quotient space and free vector space constructions, diagonalisation of hermitian matrices, algebras, homomorphisms and ideals, group of units and spectrum.
  2. Banach spaces, including dual spaces, bounded operators, bidual [and weak*-topology], completion and continuous (linear) extension.
  3. Banach algebras, including Neumann series, continuity of inversion, spectrum, C*-algebra definition.
  4. Hilbert space geometry, including Bessel's inequality, dimension, orthogonal complementation, nearest point projection for nonempty closed convex sets.
  5. Miscellaneous, including Weierstrass Approximation Theorem.
II Hilbert space and its operators (9 lectures)
  1. Sesquilinearity, orthogonal projection;
  2. Riesz-Frechet, adjoint operators, C*-property;
  3. Kernel-adjoint-range relation;
  4. Finite rank operators;
  5. Operator types: normal, unitary, selfadjoint, isometric, compact, invertible, nonnegative, uniformly positive and partially isometric;
  6. Fourier transform as unitary operator;
  7. Hardy space;
  8. Invertibility criteria;
  9. Key examples of operators, finding their spectra (shifts and multiplication operators), norm and spectrum for a selfadjoint;
  10. continuous functional calculus for selfadjoint operators, with key examples: square-root and positive/negative parts.
III Further topics (6 lectures)
  1. Polar decomposition;
  2. Matrices of operators, positivity in B(h+k), operator space - definition and simple examples;
  3. Nonnegative definite kernels, Kolmogorov decomposition;
  4. Tensor products;
  5. Hilbert-Schmidt operators;
  6. Topologies on spaces of operators (WOT, SOT, uw);
  7. Compact and trace class operators, duality;
  8. Double Commutant Theorem;
  9. Dilation and von Neumann's inequality;
  10. Two projections in general position.
Coverage of topics beyond 1-4 in III will depend on availability of time.

Lecturer


Martin Lindsay
Email j.m.lindsay@lancaster.ac.uk
Phone (01524) 594532
Photo of Martin Lindsay


Students


Photo of Heather Burgess
Heather Burgess
(Exeter)
Photo of Samuel Durugo
Samuel Durugo
(Loughborough)
Photo of Aiman Elragig
Aiman Elragig
(Exeter)
Photo of Betty Fyn-Sydney
Betty Fyn-Sydney
(Birmingham)
Photo of Andrew Hawkins
Andrew Hawkins
(Nottingham)
Photo of Anton Izosimov
Anton Izosimov
(Loughborough)
Photo of Paul  Jones
Paul Jones
(Loughborough)
Photo of Avais Kasim Sait
Avais Kasim Sait
(Lancaster)
Photo of Ana Khukhro
Ana Khukhro
(Southampton)
Photo of Antony Lee
Antony Lee
(Nottingham)
Photo of Ye Luo
Ye Luo
(Loughborough)
Photo of Yue  Wu
Yue Wu
(Loughborough)


Bibliography


Analysis nowPedersen
Introduction to Topology and Modern AnalysisSimmons
A Hilbert space problem bookHalmos


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Assessment



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