## Functional Analysis Seminar 2006-2007

Meetings are scheduled the following Thursdays at 13.45 hr in room 405 of the Snellius building:
February 15, 22
March 1, 8, 15, 22, 29
April 19, 26
May 10, 24
Officially there is a meeting at April 12, but this coincides with the Dutch Mathematical Congress of which I happen to be one of the organizers this year. So this meeting can not take place, and by the end of March we decide what we will do: cancel it or pick another day around April 12.
If necessary (and possible for the participants!) we can continue a bit following May 24, we will see about that in due time.

Passing the course and thus obtaining 6 EC happens by handing in the solutions to exercises from the book (with an average grade >= 5.5). There are 4 assignments of exercises. Earning 4 extra EC is also possible by studying extra material, see below.

Here is the reading guide to the book.

Homework assignments

First assignment (Chapters 1-2): due March 29

Chapter 1:
19: this is just linear algebra: the sequences can be any sequences and need not be in l_2.
39 and 40
74: also answer the analogue of 73.c: if a biorthogonal system exists, how can one then parameterize all biorthogonal systems in terms of a given existing one?

Chapter 2:
4: the coefficients in the summation are to be read as a_{j,j-k}. It is better to reformulate the exercise slightly and require to prove that, for a matrix (a_{jk}) with the property as given in the exercise, there exists a bounded operator A which has precisely this matrix as its matrix representation and with an estimate for its norm as given. Reason: one cannot simply write down a matrix and define A as the corresponding operator, as the relevant series need not even converge.
16
21
48: if an infinite matrix has only 1's on one antidiagonal and zeroes elsewhere, does there exist a corresponding bounded operator? Of what norm?
85: can you see eigenvectors for W*?
86: the \alpha in b should be a \signa. If you are familiar with Fourier analysis then this exercise is rather trivial. If you are not, then for b you may want to use 85.

Second assignment (Chapters 3-5): due April 19

Chapter 3:
1
9
13a: here it is meant to require to determine a one-sided inverse.

Chapter 4:
A series of related exercises (preparing for Exercise 4 in Chapter 5):
21
23: you can use the result of Exercise 21 here, as well as the fact that \lambda_j(A*A)=\lambda_j(AA*) for a compact operator A (a restatement of Corollary X.4.3).
26
27: think of 23 and 26.

Chapter 5:
4: This goes back to Hermann Weyl. You can use the result of Exercise 27 of Chapter 4 here. Of course L_2[\eta,\xi] can be identified with a subspace of L_2[0,1]; is there a natural projection around?
14: be careful about your argumentation when determining the eigenvalues (see page 172/173 for inspiration).
EXTRA QUESTION: if one applies Theorem 1.2 and Theorem 4.1 to the kernel in Exercise 14, which two identities does one obtain?

The answers to the extra question for Exercise 14 show that even very simple kernels can give nontrivial identities. If you are interested (this is not required) you may try your hand on the kernel in Exercise IV.2.a on page 188 and see what Theorem V.1.2 yields in this case. My own result was the following. Let {z_i}_{i=1}^\infty be the roots in the complex plane of the equation \pi z = cotanh (\pi z). Then there are two real roots, countably many imaginary roots and no others. The sum of the z_i^{-4} is equal to (8 \pi^4 -2)/6. Hopefully there is no mistake in the calculation...

Third assignment (Chapters 6-12): due May 10

Chapter 6:
9: the domain consists of all smooth functions vanishing at infinity.
11

Chapter 8:
3a

Chapter 10:
7
28
31 in a modified form: give an example of a self adjoint compact operator which is trace class, an example of one which is Hilbert-Schmidt but not trace class, and an example of one which is not Hilbert-Schmidt.

Chapter 11:
23
29

Chapter 12:
4: you don't need that X and Y are Banach spaces here. My own solution is very short and uses some general results which are not in this chapter, but which are at your disposal if you have taken the national course. Such a solution is of course OK; perhaps another short solution (one by first principles) is also possible.
13: you can use the polynomial spectral mapping theorem here, i.e., the result of exercise 12.

Fourth and final assignment (Chapters 13-17): due June 14

Chapter 13:
7
8: to prove that I-K is invertible you can use results from previous or future chapters if you want. To prove that I-(I-K)^{-1} is compact, the easiest way is to rewrite this algebraically. Alternatively, you can consider the function which sends a complex number \lambda in the resolvent set of K to (\lambda I - K)^{-1} - \lambda^{-1}I. Can you see that this is a compact operator for large \lambda? Combine this observation with Theorem 12.10.1 and a Hahn-Banach theorem to conclude that it is a compact operator for each \lambda in the resolvent set of K. Although the algebraic solution is the easiest one by far, the idea of using analytic functions together with a Hahn-Banach theorem is something to remember.
9

Chapter 14:
7: this shows that the notion of a Poincare operator is base dependent.
9: g_k(s) should read g_j(s).

Chapter 15:
1: in e you are only required to calculate the spectrum of U, as the rest follows by translation. Please note: in a the terms -I should be disregarded.

Chapter 16:
1a and 1b, in a reduced form: in 1a you are not required to compute the kernel and in 1b you are not required to write down the inverse.
14

Assignment for 4 extra EC: due September 3

Here is the reading guide and list of exercises for the extra material from Young's book. Don't take this on unless you are prepared to really digest this material. Studying the theory and solving the exercises will take you more than a couple of days: two weeks is more realistic. The precise data of the book are:
N. Young: An introduction to Hilbert space
Cambridge University Press, 1988
ISBN-10: 0521330718 (hardback) or 0521337178 (paperback).
The paperback version is still in print, the hardback is not. The paperback also has an ISBN-13: 9780521337175.

Week 1: Feb 8 - Feb 15
Chapter 1 entirely.

Week 2: Feb 15 - Feb 22
Chapter 2: Sections 2.1-2.15.

Week 3: Feb 22 - Mar 1
Chapter 2: first portion of Sections 2.16-2.20.

Week 4: Mar 1 - Mar 8
Chapter 2: remaining portions of Sections 2.16-2.20.

Week 5: Mar 8 - Mar 15
Chapter 3: Sections 3.1-3.4 (we will skip 3.5 and 3.6).

Week 6: Mar 15 - Mar 22
Chapters 4 and 5.

Week 7: Mar 22 - Mar 29
Chapters 6 and (if you wish, this is optional) 7.

Week 8: Mar 29 - Apr 5
Chapters 8 and 9.

Week 9: Apr 5 - Apr 12
Chapters 10 and 11.

Week 10: Apr 12 - Apr 19
Chapter 12.

Week 11: Apr 19 - Apr 26
Chapter 13.

Week 12: Apr 26 - May 3
Chapter 14.

Week 13: May 3 - May 10
Chapter 15 and part of 16.

Week 14: May 10 - May 17
Chapter 16 remainder (not 16.5); Chapter 17.