Mastermath Functional Analysis

Homework 3

I. Let B[0, 1] be the Banach space of bounded complex functions on [0, 1] endowed withthe supremum norm. For q ∈ B[0, 1] define the multiplication operator Tq : B[0, 1] →B[0, 1] by

(Tqf)(t) := q(t)f(t), t ∈ [0, 1].

(a) Show that σ(Tq) = {q(t) : t ∈ [0, 1]}, for each q ∈ B[0, 1].

(b) Determine all q in B[0, 1] for which Tq is compact.

II. Let X be a Banach space over C, let T ∈ B(X) be compact, let λ ∈ C \ {0} andconsider the equation

(T − λI)x = y. (1)

Prove the following version of the Fredholm alternative: either equation (1) has anonzero solution x for y = 0 or equation (1) has a unique solution x for each y ∈ X.

III. Consider the delay differential equation

x′(t) = −x(t− 1), t > 0x(t) = ϕ(t), t ∈ [−1, 0],

(2)

where ϕ ∈ C[−1, 0] is the initial condition. Define T : C[0, 1] → C[0, 1] by

(Tψ)(t) := ψ(1)−∫ t

0

ψ(s) ds, t ∈ [0, 1].

Here C[0, 1] denotes the Banach space of complex valued continuous functions on[0, 1] endowed with the supremum norm.

(a) If x : [−1,∞) → C is continuous, differentiable on (0,∞) and satisfies (2), thenfor every n ∈ {0, 1, 2, 3, . . .} and t ∈ [0, 1],

x(n+ t) = (T (s 7→ x(n+ s− 1)))(t).

Verify this.

(b) Prove that T is compact. (Hint: Ascoli’s theorem A5 on p. 394 may help.)

(c) Show that σ(T ) = {λ ∈ C \ {0} : e−1λ = λ} ∪ {0}.

(d) It is elementary but tedious to show that ez = z has no solution z ∈ C with|z| ≤ 1. You may use this without proof. Show that the spectral radius ρ(T ) ofT satisfies ρ(T ) < 1.

(e) Show that for each ϕ ∈ C[−1, 0] and each x : [−1,∞) → C satisfying (2) onehas x(t) → 0 as t→∞.

Furthermore, from Chapter 10 of Rudin’s Functional Analysis:

1. (xy and yx have the same spectral radius)

4. (relation between spectrum of xy and spectrum of yx)

11. (topological divisors of 0)

— Due: October 30, 2012 —

Teaching Assistant: Bjorn de Rijk, brijk@math.leidenuniv.nlIf you hand it in by email, send it as one PDF.

Leiden UniversityMarcel de Jeu

Onno van Gaans