SOLUTIONS FOR HOMEWORK 3
Chapter VIII
10. (a) For x D(A),
(A + 1)x
2
= (A + 1)x, (A + 1)x
= Ax, Ax + Ax, x + x, Ax + x, x Ax
2
+ x 2.
(b) Suppose (yn ) is a Cauchy sequence in ran(A + 1), and show that its limit y
belongs to ran(A + 1) as well. Write yn =
HOMEWORK 5
Problems assigned up to and including Friday, 05/04. The solutions are due
on Monday, 05/14.
Assigned Wednesday, 05/02:
In the following two problems, E is a projection-valued measure on Borel
subsets of R. That is, for any Borel R, E() L(H) is
SOLUTIONS FOR HOMEWORK 2
Chapter VIII
4. (a) As C is symmetric, the inequality (C + i)x x holds for every
x D(C). In particular, C + i is injective. If C = A, then D(A)
D(C),
hence ran(A + i) ran(C + i).
(b) This statement follows from Exercise A. For the
SOLUTIONS FOR HOMEWORK 1
Chapter VIII
1. T e = e , hence (e , e ) (T ). To prove (e , 0) (T ), write
e = i i , with i |i |2 = e 2 (by assumption, innitely many i s
i=1
are non-zero). For N N, let eN = N i i . Then limN e eN = 0. By
i=1
denition, T (eN ) =
HOMEWORK 6
Problems assigned up to and including Monday, 05/14. The solutions are
due on Friday, 05/25.
Assigned Friday, 05/11:
VIII.39. Ignore the instructions in (a) and (b): it suces to prove the selfadjointness of A by one method, of your choice. You
HOMEWORK 4
Problems assigned up to and including Friday, 04/27. The solutions are due
on Monday, 05/04.
Assigned Wednesday, 04/25:
A. Consider the Hilbert space H = L2 (), where is a -nite measure.
We say that a symmetric operator T on H is real if (i) x
HOMEWORK 3
Problems assigned up to and including Friday, 04/20. The solutions are due
on Friday, 04/27.
Assigned Wednesday, 04/18:
Chapter VIII: 10, 14.
A. Suppose T is a closed densely dened operator on a Hilbert space H. Prove
that ker T = cfw_x H : T x
SOLUTIONS FOR HOMEWORK 6
VIII.39. Ignore the instructions in (a) and (b): it suces to prove the selfadjointness of A by one method, of your choice. You can rely on the material
from Laxs book.
Following the notation of [P. Lax, Functional Analysis, Sectio
SOLUTIONS FOR HOMEWORK 5
In the following two problems, E is a projection-valued measure on Borel
subsets of R. That is, for any Borel R, E() L(H) is a projection,
and certain conditions are satised. Furthermore, the operator T is dened
via T = t dEt , wi
HOMEWORK 1
Problems assigned up to and including Friday, 04/06. The solutions are due
on Friday, 04/13.
Assigned Wednesday, 04/04:
Chapter VIII: 1, 2.
Assigned Friday, 04/06:
Chapter VIII: 3.
A. Suppose A : D(A) H is a self-adjoint operator with trivial k
HOMEWORK 2
Problems assigned up to and including Friday, 04/13. The solutions are due
on Friday, 04/20.
Assigned Wednesday, 04/11:
Chapter VIII: 4, 5.
A. Suppose T is a symmetric operator. Dene its deciency indices n+ =
codim ran(T i) and n = codim ran(T
FINAL EXAM: DUE MONDAY, JUNE 11, 1PM
VIII.44. Hint. Problems A and B below may be helpful. You have to assume
T is densely dened otherwise, T is not dened.
A. Suppose T is a densely dened operator on a Hilbert space H, and A
L(H). Prove that (AT ) = T A
SOLUTIONS FOR HOMEWORK 4
A. Consider the Hilbert space H = L2 (), where is a -nite measure.
We say that a symmetric operator T on H is real if (i) x D(T ) whenever
x D(T ), and (ii) T x is real-valued whenever x is real-valued. Here, x denotes
the complex