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please provide detailed answer for the question. The answer you provided before was wrong.
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please provide detailed answer for the question. The answer you provided before was wrong.
please provide detailed answer for the question. The answer you provided before was wrong.
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+E,y+n] and (a +iB) [x, y] := [ax - By, ay + Bx] ([x, y], [§, 7) He + (y, n) H for x, y, E, n H and a, ß E R. The inner product on He is defined by =(x, E) Hi(x, n) + (y, c) Moreover, the complexification Te of T is given by Tc[x, y] for all x, y, E, ne H. H = [Tx, Ty] for all x, y = H. Then, (Hc, (,) c) i is a complex vector space with inner product and Te is a linear operator on Hc. (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 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 € H. (e) Assume that T is compact, self-adjoint, and not the zero operator. Conclude that there exist an orthonormal system (ek) EM in H indexed by MCN and real numbers ER\ {0} for ke M such that Ta = Σ με· (2,ekmek KEM for all x EH.
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+E,y+n] and (a +iB) [x, y] := [ax - By, ay + Bx] ([x, y], [§, 7) He + (y, n) H for x, y, E, n H and a, ß E R. The inner product on He is defined by =(x, E) Hi(x, n) + (y, c) Moreover, the complexification Te of T is given by Tc[x, y] for all x, y, E, ne H. H = [Tx, Ty] for all x, y = H. Then, (Hc, (,) c) i is a complex vector space with inner product and Te is a linear operator on Hc. (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 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 € H. (e) Assume that T is compact, self-adjoint, and not the zero operator. Conclude that there exist an orthonormal system (ek) EM in H indexed by MCN and real numbers ER\ {0} for ke M such that Ta = Σ με· (2,ekmek KEM for all x EH.