Your conclusion is correct but I'm not sure I agree with your proof. In particular this
(B) = λ'B
is only true if B is an eigenvector of T, and thats not really clear a priori. Just say that B is an eigenvector of T so
T(B) = λB
to both sides
(T(B)) = T-1
T is a linear operator (I assume) ... I'll let you finish the argument. It's pretty trivial.
Also as to the diagonalization comment, while I agree that's a good way to think about it intuitively it's not a good technique for a proof. Obviously there are issues with some finite dimensional space maps being non-diagonalizible; but even more fundamentally for a Hilbert space I think you would need the operator to be compact to have any hope of such an approach working.