Description

Linear Operators
Assignment 9
Exercise 9.1
Let M := {u ∈ L2([0,1]): u ∈ C2(0,1), u(0) = u(1) = 0} and define

on M ⊂ L2([0,1]). Let K: L2([0,1]) → L2([0,1]) be given by
1
(Ku)(x) :=g(x,ξ)u(ξ)dξ
0
with
{x(1−− ξ) x < ξ,≥
g(x,ξ) := ξ(1 x) x ξ.
It is known that K is compact and self-adjoint. Furthermore, K = L−1.
i) Show that L = K−1 if K is restricted to M. (You will have to perform some careful differentiations; use the chain rule.) (2Marks)
ii) Consider the eigenvalue equation Lu = λu for u ∈ M. Show that λ = 0̸ is an eigenvalue for L if and only if 1/λ is an eigenvalue for K. How are the eigenfunctions of K and L related? (2Marks)
iii) Find the eigenvalues λn∈C and eigenfunctions ψn∈ M of L by solving an ordinary differential equation. You will have to prove first that λn∈R and then consider the cases λn > 0, λn = 0 and λn < 0. (5Marks)
iv) Find the spectrum of K. Is zero an eigenvalue? Find the sequence of eigenvalues (λn) such that λn↘ 0. (3Marks)
v) According to the spectral theorem, L2([0,1]) has an orthonormal basis consisting of eigenfunctions of K. Use this to deduce that a certain Fourier basis is actually a basis.
(2Marks)