All this doesn't make much sense: you're going to define the annihilator of S^o OR you're going to define the annihilator S^o nof some set S?
I'm almost sure it must be the latter. Now, what is V*? the v.s. of all linear functionals on V?
later on you talk of N(T^t)...what is this? The nullity or kernel of T^t?
Be more careful and if you want write again your question.