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MATH 611 (Spring 2009) Homework #9 Due April 28th
(1) Let H be a Hilbert space and y, z ∈ H. Define T ∈ B(H) by T (x) = (x, y)z. Show that T is
compact.
(2) (a) Let H be an infinite dimensional Hilbert space with an orthonormal basis {un } and let
T ∈ B(H). Show that if T is compact then kT (un )k → 0 as n → ∞. [Hint:
PSuppose kT (un )k 9 0.
Choose a subsequence un(m) so that kT (un(m) ) − yk < 1/m and show xk = km=1 m1 un(m) is bounded
but T (xk ) has no convergent subsequence.]
∞
∞
P
P
∞
(b) Hence show that if T is compact and T (
x n un ) =
λn xn un , where {λ}∞
n=1 ∈ ` , then
n=1
λn → 0 as n → ∞.
n=1
(3) Let {en } and {fn } be orthonormal bases of an infinite dimensional separable Hilbert space
H, and let T ∈ B(H). Prove that
(a) the condition for an operator to be Hilbert-Schmidt does not depend on the choice of orthonormal basis for H by showing that
∞
X
n=1
2
kT en k =
∞
X
∗
2
kT fn k =
∞
X
kT fn k2 .
n=1
n=1
(b) T is Hilbert-Schmidt iff T ∗ is Hilbert Schmidt.
(4) Let H be an infinite dimensional separable Hilbert space and S, T ∈ B(H). Prove that
(a) If S or T is Hilbert-Schmidt, then S ◦ T is Hilbert-Schmidt. [Hint: In the case that S is HilbertSchmidt use (ST )∗ = T ∗ S ∗ and 3(b).]
(b) If T has finite rank, then it is Hilbert-Schmidt. [Hint: Construct an orthonormal basis of H
consisting of elements of R(T ) and R(T )⊥ and use (3) above.]
(5) Let {en } and {fn } be orthonormal bases of an infinite dimensional separable Hilbert space
H. Let {αn } be a sequence in C. Define T : H → H by
Tx =
∞
X
αn (x, en )fn .
n=1
Prove that
(a) T is bounded iff αn is bounded.
(b) T is compact iff αn → 0. [Hint: See (2a) above and Proposition 15.8.]
∞
P
(c) T is Hilbert-Schmidt iff
|αn |2 < ∞;
n=1
(d) T has finite rank iff ∃N so that αn = 0 for all n > N .
It follows that not all compact operators are Hilbert-Schmidt.
Additional Questions (Not to be handed in)
(6) Prove that the set of compact operators is a vector subspace of the space of bounded operators.
(7) Prove that the multiplication operator Tf (g) = f g on L2 [a, b] is not compact if f is a nonzero
continuous function. [Hint: Use (1)]
(8) Prove Example 15.11 from class and do [F, 6.3.4, 6.3.5]