My book last year gave this as a problem. It says to prove that N(T) = N(T*), and R(T) = R(T*). It gives a hint: use theorem 6.15 and excercise 12. Theorem 6.15 is just the basic facts about normal operators:
|Tx| = |T*x|
T - cI is normal for all c in F
If Tx = cx, then T*x = c*x, where c* denotes the conjugate of c.
If c and d are distinct eigenvalues of T with corresponding eigenvectors x and y, then x and y are orthogonal
Exercise 12 says to prove, for any linear operator T on any inner product space V
a) the orthogonal complement of R(T*) is N(T)
b) If V is finite dimensional, then R(T*) is the orthogonal complement of N(T)
It gives a hint to use another exercise which says:
V is an inner product space, S and S' are subset of V, W is a finite dimensioanl subspace of V. Prove:
a) S' contained in S implies the orthogonal complement of S is contained in the orthogonal complement of S'
b) S is contained in the orth. compl. of the orth. compl. of S, so span(S) is containd in the orth. compl. of the orth. compl. of S
c) W = the orth. compl. of the orth. compl. of W
d) V = W direct sum with the orth. compl. of W
For c, use the fact that if W is a f.d. subspace of an i.p.s V, and y is in V, then there exist unique u in W and z in orth. compl. of W such that y = u + z. For d, prove the fact that if W and W' are subspaces of a vector space V, then V is their direct sum iff for all v in V, there exist unique w and w' in W and W' respectively such that v = w + w'.
