Description

Linear Operators
Assignment 7
Exercise 7.1
Consider again the left- and right-shift operators L and R on ℓ2. The goal of this exercise is to show directly that the numbers λ ∈ C with |λ| = 1 lie in the continuous spectrum of these operators.
An eigenvector for the left-shift operator satisfying Leλ = λ · eλ is given by
eλ = (1,λ,λ2,λ3,…).
However, for |λ| = 1 this vector is not in ℓ2. Consider instead the “almost-eigenvector”

for |λ| = 1.
i) Show that .
(0.5Marks)
ii) Calculateand show that.
(1Mark)
iii) Deduce that λ ∈ σcontinuous(L). (0.5Marks) iv) Find a sequence of almost-eigenvectors for the right-shift operator R and show that all λ ∈ C with |λ| = 1 lie in σcontinuous(R).
(2Marks)
Exercise 7.2
Let L: L2([0,1]) → L2([0,1]) be given by
(Lu)(x) = x · u(x)
i) Show that the domain of the inverse is dense. Hint: the domain surely includes the set
.
and you can show that M is dense by hand. (2Marks)
ii) Show that L is bounded, find ∥L∥ and verify that L−1 is unbounded. (3Marks)
iii) Find the state of L and of L−1. (2Marks)
iv) Is L self-adjoint? (1Mark)
v) Find the spectrum of L. (3Marks)
Exercise 7.3
On L2([0,1]) consider the operator L defined by

i) Show that the state of L is (II,1n). Hint: does the range of L contain the set of polynomials? (2Marks)
ii) Find the adjoint of L.
(1Mark)
Exercise 7.4
For two sequences (an),(bn) ∈ ℓ1 the Cauchy product of their series is defined as

where
.
(It can be shown that the equality holds and that (cn) ∈ ℓ1.) The sequence (cn) is said to be the convolution of (an) and (bn) and we write
(cn) = (an) ∗ (bn).
Prove Young’s convolution inequality:
∥(an) ∗ (bn)∥p ≤ ∥(an)∥p · ∥(bn)∥1 1 ≤ p < ∞.
Instructions: for the case p > 1, write

and apply Hölder’s inequality. (3Marks)