Answer Happy

4. Let A∈B(H) be self-adjoint. (a) Prove part vi) of Theorem 8.33, whose proof was skipped in the lecture: if Av=λv, then also f(A)v=f(λ)v for all f∈B(R). (b) Does f(σ(A))=σ(f(A)) hold for all f∈B(R) ? Hint: Consider the multiplication operator A∈B(L2[0,1]),(Ag)(x):=xg(x).