Answer Happy

Exercise 10.1: Spectral theorem for operators on a real Hilbert space (1+1+1+3+2=8 points) Consider a Hilbert space H over R with inner product (,) and an operator TEL(H). Define the so-called complexification of H as Hc = H x H, where we use square brackets to signify an element of this space. Addition and scalar multiplication in Hc are defined by [x+E,y+n] and (a +iB) [x, y] := [xx- -By, ay + Bx] [x, y] + [E, n] for x, y, E, n H and a, ß E R. The inner product on He is defined by ([x, y], [E, n]) He =(x, E)-i(x, n) + (y, E) + (y,n)H H := for all x, y, E, ne H. Moreover, the complexification Tc of T is given by Te[x, y] = [Tx, Ty] for all x, y € H. Then, (Hc,(,)Hồ) is a complex vector space with inner product and Te is a linear operator Hc. = (a) Show that [x, y] = - ||*|| + || || holds for all æ, y € H. Conclude that (Hc, (,·•) H.) is a Hilbert space and Te E L(HC). (b) Show that (Tc)* = (T)c. Conclude that if T is normal (respectively self-adjoint), then Te is normal (respectively self-adjoint). (c) Prove that if T is compact, then Te is compact. (d) Assume that T is compact and self-adjoint. Show that for every μ € op (Tc) {0} exists an orthonormal basis of N(μld - Tc) such that every basis vector is of the form [e, 0] for some e EH. (e) Assume that T is compact, self-adjoint, and not the zero operator. Conclude that there exist an orthonormal system (e)kEM in H indexed by MCN and real numbers k ER {0} for kEM such that Ta = Σ μ · (x,ek) Hek for all x EH. KEM