MATHEMATICS 421/510 HOMEWORK 5
Due Friday, April 8
Marking: Since this is the end of term :), everyone gets 40% for free. For the remaining
60%, you need to get 3 problems out of the 4 below (if you do all 4 then I will drop the
lowest score).
2
∞
1. Let {φj }∞
1 be an orthonormal basis for L ([0, 1]). Prove that {φj (x)φk (y)}i,j=1 is an
orthonormal basis for L2 ([0, 1]2 ). (Hint: To prove completeness, it suffices to prove that
Parseval’s identity holds.)
2. Prove that if T is a compact operator on a separable Hilbert space H, then there is a
sequence of finite-rank operators Tn on H such that kTn − T k → 0.
(More generally, the conclusion of (a) holds for compact operators on a Banach space X ,
provided that X has a Schauder basis, i.e. a countable linearly
P independent set {ej } ⊂ X
such that every x ∈ X can be uniquely represented as x = j xj ej for some sequence of
scalars {xj }. It may fail if X does not have a Schauder basis. This is for your information
only – you don’t have to prove it.)
3. Find the spectrum of the operator T on H = L2 ([0, 2]), defined by
(
(T f )(x) =
2xf (x), 0 ≤ x ≤ 1,
2f (x), 1 ≤ x ≤ 2.
4. Let T be a compact symmetric operator on a Hilbert space H. Let {ej } be an orthonormal basis for H consisting of eigenvectors of T , with the corresponding eigenvalues {λj }
(i.e. T ej = λj ej ).
(a) If P (x) = an xn + . . . + a1 x + a0 is a polynomial with coefficients aj ∈ R, define
P (T ) = an T n + . . . + a1 T + a0 I (T j means T composed with itself j times). Prove that
P (T ) is symmetric, and that for all x ∈ H we have
P (T )x =
X
P (λj )hx, ej iej .
j
(b) Assume that λj ≥ 0 for all j. Prove that there is a compact symmetric operator S
such that T = S 2 .
References: In preparing this part of the course, I used the following books:
• Functional Analysis: An Introduction, by Y. Eidelman, V. Milman, and A. Tsolomitis,
AMS 2004;
• A Course in Functional Analysis, by J. B. Conway, Springer 1989.
There are many other good textbooks in functional analysis, just browse the Math library.
The material we have covered is standard and should be easy to find, except that there
may be slight differences in terminology (so always check the definitions!).
1