I am trying to prove or to break the following statement (I assume that the statment is correct):

**Assumptions:** Let $H$ be a Hilbert-space (or more generally a reflexive space) and $T\in \mathcal{L}(H)$ an operator with the additional property that $\frac{1}{n}T^n$ converges to zero in the strong operator topology on $\mathcal{L}(H)$.

**Assertion**: The dual operator $\frac{1}{n}{T'}^n$ converges to zero in the strong operator topology on $\mathcal{L}(H')$.

I am thankful for any piece of advice, references or counterexamples.