Exercise 10.1: Spectral theorem for operators on a real Hilbert space (1+1+1+3+2=8 points) Consider a Hilbert space H ov

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Exercise 10 1 Spectral Theorem For Operators On A Real Hilbert Space 1 1 1 3 2 8 Points Consider A Hilbert Space H Ov 1 (106.46 KiB) Viewed 8 times

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 He = 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, y] + [E, n] = [x+, y + n] for x, y, E, n H and a, 3 € R. The inner product on He is defined by ([x, y], [§, 1)) He =(x, E) - (x,n) H +i(y, E) H+ (y, n) H for all x, y, E, n € H. Moreover, the complexification Te of T is given by Te[x, y] [Tx, Ty] for all x, y € H. Then, = (Hc,(,)n) is a complex vector space with inner product and Te is a linear operator on Hc. and (a +iB) [x, y] [ax - By, ay+ Bx] (a) Show that ||[*,y]|| = ||+| holds for all x, y € H. Conclude that (Hc, (,) Hc) is a Hilbert space and Te E L(Hc). (b) Show that (Tc) = (T)c. Conclude that if I 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 μE 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 € H. (e) Assume that T is compact, self-adjoint, and not the zero operator. Conclude that there exist an orthonormal system (ek)keM in H indexed by MCN and real numbers ER\ {0} for k € M such that Tx = Σ με · (w,ekmek KEM for all x EH.