Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

Question

I was recently met with this in my functional analysis class on which I am stuck:

Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \leq 1 $ in the operator norm) . We are given these three conditions on T:

1. $ \forall h \in \mathbb{H} $ we have the convergence in norm $ T^n \to 0 $.

2. $ \forall h \in \mathbb{H} $ we have the convergence in norm $ T^{*n} \to 0 $.

And we are asked to determine which (if any) of these conditions necessarily leads to the property that for all $ h \in \mathbb{H} $ we have $ ||Th || < ||h|| $ ?

Now I do not really know how exactly to approach this as I only know the operator norm of an operator and its adjoint are equal but nothing else. I tried giving it a go but got nothing. I would certainly appreciate the much needed help on this. Thanks all helpers.