MATH 675 – HOMEWORK 8
DUE 7 DECEMBER 2006
Exercise 1. Prove that a finite rank operator on a Hilbert space is compact. Recall that a
A∈ L(H, H ) has finite rank if ran(A) is finite dimensional.
Exercise 2.
(a) Prove that if {ϕn}∞
n=1 is an orthonormal system in the Hilbert space H, and if n6=m,
then kϕn−ϕmk=√2.
(b) Prove that the identity operator on an infinite-dimensional Hilbert space His not
compact.
Exercise 3. Kolmogorov, p. 251, Problem 2. (The first part only, i.e., prove that Ais a
compact operator, or in Kolmogorov’s words, a completely continuous operator.)
Exercise 4. Prove that if A∈ L(H, H ) is compact, and B∈ L(H, H) is arbitrary, then
AB ∈ L(H, H ) is compact.
Exercise 5. Prove the following facts about orthogonal projectors. Assume that Pis the
orthogonal projector onto the closed subspace Mof a Hilbert space H.
(1) P2=P. An operator which satisfies this criterion is said to be idempotent.
(2) I−Pis the orthogonal projector onto M⊥.
(3) Pis a bounded linear transformation on Hwith kP xk ≤ kxk, and in fact, kPk= 1.