For the first part D is the differentiation transformation where f -> f'. It is anti0Hermitian on V st D*=-D which implies that the operator T is non-negative or >= 0 for each basis vector. So D*D = -DoD = D^2 = T. I need help with a-c, I know T is diagonal so ?T should be able to be computed.

Prob 3. Show that the operator T = -D2 is nonnegative on the space V := span(1, cosx, sin x) over R, with the inner product (f, g) : = f(x)g(x)dx. Find (a) its square root operator VT; (b) an example of a self-adjoint operator R # VT such that R2 = T; (c) an example of a non-self-adjoint operator S such that S*S = T