Prob 4. Show that the operator T = D2 is nonnegative on the space V :=span(1, cos cc,sin:c) over IR, with the inner product (f, g) :f(x)g(m)dm. Find…

Let T1 and T2 be normal operators on an n-dimensional inner product space V . Suppose both have n distinct eigenvalues ?1, . . . , ?n. Show that there is an isometry S ? L(V ) such that T1 = S?T2S. Attachment 1Attachment 2Prob 4. Show that the operator T = —D2 is nonnegative on the space V :=span(1, cos cc,sin:c) over IR,with the inner product (f, g) := :f(x)g(m)dm.Find(a) its square root operator x/i(b) an example of a self—adjoint operator R 7E x/T such that R2 2 T; (c) an example of a non—self—adjoint operator 8 such that 8*8′ = T.

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