Graduate Course in Analysis, Autumn Term 2001

This course is in four parts, run jointly between York and Leeds, and is aimed at first year graduate students, although of course anyone else is also welcome to attend.

1. Weeks 1-3 in York Simon Eveson A brief introduction to Banach spaces, their duals, and L^p-spaces
   Wednesday 10 and 17 October, 14:00-15:30 in G/010, Wednesday 24 October 11:30-13:00 in V/131.

   • definition and examples of normed and Banach spaces, including l^p and L^p;
   • a brief and largely proof-free introduction to measure theory;
   • dual spaces and L^p-L^q duality;
   • the Hahn-Banach Theorem and some immediate consequences;
   • definition and examples of reflexive spaces;
   • some further applications of the Hahn-Banach Theorem.

   Literature: Rudin Real and complex analysis, Hewitt and Stromberg Real and Abstract Analysis, Diestel Sequences and series in Banach spaces.

   Notes for this section are available in PostScript. There is also a list of errata for the printed version (the PostScript file linked to above has been corrected).

2. Weeks 4-5 in York Sandra Pott Harmonic functions and H^p spaces.
   Wednesday 31 October 11:30-13:00 in V/131, Wednesday 7 November 14:30-16:00 in G/010.

   • Harmonic functions, basic properties;
   • Poisson integral;
   • L^p convergence;
   • Hardy-Littlewood maximal function;
   • non-tangential limits;
   • subharmonic functions;
   • H^p spaces.

   Literature: Rudin Real and complex analysis, Garnett Bounded analytic functions, Douglas Banach algebra techniques in operator theory, Young An introduction to Hilbert space.

   Notes for this section are available in PostScript.

3. Weeks 6-7 in Leeds Garth Dales Banach Algebras
   Tuesday 13 and 20 November 11:00-12:00 in Classroom D and 14:00-15:00 in Tutorial Room 4. IMPORTANT: 20th November lecture CANCELLED.

   • Basic Properties;
   • ideals;
   • characters;
   • maximal ideal space;
   • Gelfand transform.

4. Weeks 8-9 in Leeds Jonathan Partington
   Tuesday 27 November and Tuesday 4 December 13:00-15:00 in Tutorial Room 4.

   • inner-outer factorization of H^p functions;
   • Beurling's Theorem;
   • Wiener's Theorem;
   • Toeplitz operators;
   • Hankel operators;
   • Nehari's Theorem.

   Literature: Douglas Banach Algebra Techniques in Operator Theory, Nikolski Treatise on the Shift Operator.

5. Week 10 in Leeds Vladimir Kisil
   Tuesday 11 December 13:00-15:00 in Tutorial Room 4.

   More Toeplitz operators, localisation.