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 = maxfjλj : λ eigenvalue of Ag for any operator on a finite- dimensional Hilbert space? (2) Let A be a normal operator such that σ(A) = {−3g [ [0; 1] [ f2g. Compute kA11 − 2A2k. (3) Let fλ1; : : : ; λrg ⊆ 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 2 ρ(T ). b) Give a characterization of the normality/unitarity/self-adjointness/positivity of A in terms of the numbers fλ1; : : : ; λrg. 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 : fλ1; : : : ; λrg ! C. a) Show that f 7! f(A) is an algebra morphism from Cfλ1,...,λrg 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 2 Cfλ1,...,λrg: 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 = [n2NMn, where the Mn are pairwise A P A A disjoint, then P (M) = n2N 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 2 [0; 1=2]; φ(x) := 1; x 2 [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. 2.