Math 241A Fall 2008, HOME WORK

Please turn in ths HW the beginning of section Fri Oct 10
Please tell me what you think of the Homework.         I NEED FEEDBACK!!

Ch 1 p6 sec 1   ex 3 Show that H is a preHilbert space. You do not need to prove it is complete.
Chapter 1         p6 sec 1   ex 6         .
p11 sec 2   ex 1,2, 3
p13 sec 3   ex 5
p18 sec 4   ex 13, 19
p23 sec 5   ex 2, 3

Chapter 2 sec 1 Great Examples of Operators   ex 2
Section 2, Adjoints ex. 6, 11, 16
Turn the above in on FRI Oct 10 2008

Chapter 2
Section 1, ex. 1, 3, 5, 8
Section 3 Projections , ex. 6, 11
Section 4 Compact Operators, ex. 1, 2, 4, 5
Section 5, Diagonalizin Comapact SelfAdj Operators, ex. 1

Turn the above in on FRI Oct 24 2008

Chapter 3 Banach Spaces

Bill already lectured on sec 1, 2, 3 in the early part of the course while discussing Hilbert Space.
Section 1, ex. 3
Section 2, ex. 5, 6
Section 3, ex. 1, 2, 3
Section 4, Quotient Spaces ex. 1, 6
Section 5, Linear Functionals , ex. 2
Turn the above in on FRI Nov 7 2008

TURN IN THE FOLLOWING BY WEDS Nov 26

Section 6, HB Theorem , READ IT
Section 7, Banach Limits, p. 83
Section 9, Ordered Vector Spaces, p.88, ex. 4, 6, 7, 8, 9
Ch 4 Section 3   Ex 2, 7   and   Ex 10 with TVS replaced by Banach Space.

B1.   As in the Banach Limit section 3.7
Take V to be l^\infty   and   S = sequences which posess a limit   C= all nonnegative sequences in V.
What are (all of) the order units in S?
Say exactly which symmetric matrices have a cyclic vector

Friday Dec 5 no lecture by Bill, I suggest going to Prof Sturmfels lecture at 1PM in APM 6402

Fri Dec 12: Turn in the following (you can slip it under my office door)

Suppose A1 A2 A3 are commuting self adjoint operators on a seperable Hilbert space H.
Suppose the spectrum of Aj is contained in [-2,2] for each j.
Let R:= [-2,2]^3
The following is true and you may assume it:
If p(x) is positive on R, then p(A) is PsD. Here x =(x1,x2,x3).

PROBLEM:
1. What is a natural notion of cyclic vector for A1 A2 A3
2. Assuming that they have a cyclic vector, what is a natural spectral representation for A1 , A2, A3?
3. Prove it .......................................... .............................

Friday Dec 5 no lecture by Bill, I suggest going to Prof Sturmfels lecture at 1PM in APM 6402