A key property about adjoint operators is that the null space of T is the orthogonal complement of the range of T*. Reason:
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If n(T) is the nullity of T and r(T) is the rank of T, it follows that r(T*) = dim(V) – n(T). Therefore, by the "rank plus nullity" theorem, n(T*) = n(T).
Another key property is that the null space of T*T is the same as the null space of T. The inclusion one way round is obvious: if Tx=0 then T*Tx=0. To do it the other way round, notice that if T*Tx=0 then . Hence n(T*T) = n(T) and therefore r(T*T) = r(T).