MATH 520 Homework
Spring 2014
21. Give an example in H = C2 showing that r(T ) < ||T || can happen. (Remark:
a theorem we will show later, or that you may know from linear algebra, states that
equality will hold if the matrix of T is Hermitian).
22. (See problem 9.3 in text) Recall that the resolvent operator of T is defined to be
Rλ = (λI − T )−1
a) Prove the resolvent identity
Rλ − Rµ = (µ − λ)Rλ Rµ
λ, µ ∈ ρ(T )
b) Deduce from this that Rλ , Rµ commute.
c) Show also that T, Rλ commute for λ ∈ ρ(T ).
23. (See problem 9.10 in text) Let T denote the left shift operator on `2 . Show that
a) σc (T ) = {λ : |λ| = 1}
b) σp (T ) = {λ : |λ| < 1}
c) σr (T ) = ∅
24. If λ 6= ±1, ±i show that λ is in the resolvent set of the Fourier transform F.
(Suggestion: Assuming that a solution of Fu − λu = f exists, derive an explicit formula
for it using
F 4 u = λ4 u + λ3 f + λ2 Ff + λF 2 f + F 3 f
and the fact that F 4 = I if F is the Fourier transform.)
25. (See problem 10.13 in text) Let H = L2 (0, 1), T1 u = T2 f = T3 u = u0 on the domains
D(T1 ) = {u ∈ AC[0, 1] : u0 ∈ H}
D(T2 ) = {u ∈ AC[0, 1] : u0 ∈ H, u(0) = 0}
D(T3 ) = {u ∈ AC[0, 1] : u0 ∈ H, u(0) = u(1) = 0}
Show that
(i) σ(T1 ) = σp (T1 ) = C
(ii) σ(T2 ) = ∅
(iii) σ(T3 ) = σr (T3 ) = C.