Functional Analysis Exercise Class

Week: February 1–5

Exercises

1 2 (1) a) Let the operator A on C2 be given by the matrix A = . 0 1 Compute the spectrum, the spectral radius, and the norm of A, when C2 is equipped with the usual Euclidean inner product. b) Is it true that kAk = max{|λ| : λ eigenvalue of A} for any operator on a ﬁnite- dimensional Hilbert space? (2) Let A be a normal operator such that σ(A) = {−3} ∪ [0, 1] ∪ {2}. Compute kA11 − 2A2k.

(3) Let {λ1, . . . , λr} ⊆ C, and P1,...,Pr be non-zero orthogonal projections on a complex Hilbert Pr space that are pairwise orthogonal, i.e., PiPj = 0 when i 6= j, and, moreover, i=1 Pi = I. Pr Let A := i=1 λiPi. a) Determine the spectrum of A, and give the resolvent operator (λI − A)−1 for every λ ∈ ρ(T ). b) Give a characterization of the normality/unitarity/self-adjointness/positivity of A in terms of the numbers {λ1, . . . , λr}. c) Determine the norm of A. d) What is the spectrum of an orthogonal projection? (4) In the setting of the previous exercise, deﬁne

r X f(A) := f(λi)Pi i=1

for every function f : {λ1, . . . , λr} → C.

a) Show that f 7→ f(A) is an algebra morphism from C{λ1,...,λr} to B(H), under which the constant one function is mapped to I, the identity function id is mapped to A, and idz 7→ z¯ is mapped to A∗. Pn k Pn k b) Show that for every polynomial p(x) = k=0 ckx , p(A) = k=0 ckA , i.e., f 7→ f(A) is an extension of the polynomial function calculus. c) Prove the spectral mapping theorem

σ(f(A)) = f(σ(A)), f ∈ C{λ1,...,λr}.

A d) Calculate P (M) := 1M (A) for any M ⊆ C. These are called the spectral projections of A, corresponding to the set M. Show that if M = ∪n∈NMn, where the Mn are pairwise A P A A disjoint, then P (M) = n∈N P (Mn). Because of this σ-additivity property, P (.) is called a projection-valued measure.

1 (5) Let H = l2(N, C), and S : H → H be the right shift operator given by (Sx)n := xn+1, i.e., S(x1, x2, x3,...) = (x2, x3,...). Determine the point, continuous, and residual spectrum of S and S∗.

(6) Let L2([0, 1], C) be the completion of C([0, 1], C) with respect to the inner product hf, gi := R [0,1] f(t)g(t) dt. Let ( 2x, x ∈ [0, 1/2], φ(x) := 1, x ∈ [1/2, 1],

and let Mφ : f 7→ φf be the multiplication operator deﬁned by φ. Determine the point, continuous, and residual spectrum of Mφ.