Description

Linear Operators
Assignment 8
Exercise 8.1
Define the operator L: ℓ2 → ℓ2 by
.
i) Show that L is not a Hilbert-Schmidt operator. (1Mark)
ii) Show that L is self-adjoint. (1Mark)
iii) Show that L is compact. (2Marks)
iv) Find upper and lower bounds for the spectrum of L. (2Marks)
v) Find the spectrum of L.
(4Marks)
Exercise 8.2
Let M := {u ∈ L2([0,1]): u ∈ C2(0,1), u(0) = u(1) = 0} and
.
Let K: L2([0,1]) → L2([0,1]) be given by
1
(Ku)(x) := ∫ g(x,ξ)u(ξ)dξ
0
x < ξ, x ≥ ξ.
with
i) Show that K is the inverse of L, i.e., KL = I on M. (This requires some elementary calculations with the integral.)
(2Marks)
ii) Show that L is unbounded. (2Marks)
iii) Show that K is compact. (2Marks)
iv) Show that g(x,ξ) = g(ξ,x) and deduce that K is self-adjoint. (2Marks)
v) Find the upper and lower bounds of the Rayleigh quotient for K.
(2Marks)