Do powers of contraction on Hilbert space converging to zero imply convergence of its adjoint to zero also?

Question

In my functional analysis class I was met with the following problem:

We suppose that $ \mathbb{H} $ is a Hilbert space and that T is a contraction operator on H (meaning $ ||T|| \leq 1 $ in the operator norm) and we are given these conditions:

1. For all $ h \in \mathbb{H} $ we have $ T^nh \to 0 $ (n is a natural so of course we deal with sequences).

2. For all $ h \in \mathbb{H} $ we have $ T^{*n}h \to 0 $ (n is a natural so of course we deal with sequences).

3. For all $ h \in \mathbb{H} $ we have $ T^nh \to 0 $ and $ T^{*n}h \to 0 $.

We are asked to determine for these three statements which leads to which. We are not given the dimension of the Hilbert space $ \mathbb{H} $, and it is obvious from definition that 3 leads to both 1 and 2, but in terms of 1 leading to 2 and vice verse I got no idea or how to start. If T converges to zero then does its adjoint also? The opposite? I figured if one statement does not lead to another there is a counterexample involved of a contraction T converging to zero but its adjoint not converging to zero or vice versa. I only know that for a contraction its adjoint is a contraction as well as their norms are equal. I got no idea how to approach this so I am posting here in the hopes of receiving help. Thanks to all kind helpers.

Answer 1

Take $H=\ell^2$, and $S$ the right shift operator $S(x_1,x_2,...)=(0,x_1,x_2,...)$. It is clear that $S$ is an isometry. Therefore a contraction. Moreover, $S^*=(x_2,x_3,...)$ (the left shift operator), and $\|S^*(x_1,x_2,...)\|^2=\|(x_2,x_3,...)\|^2=\sum_{k=2}^\infty|x_k|^2\le\sum_{k=1}^\infty|x_k|^2=\|(x_1,...,)\|^2$, so $S^*$ is also a contraction.

Now, $$\begin{array}{} \|S^n(x_1,x_2,...)\|^2&=&\|(0,0,...,0,x_1,x_2,...)\|^2&=&\|(x_1,x_2,...)\|^2&&\\ \|S^{*n}(x_1,x_2,...)\|^2&=&\|(x_{n+1},x_{n+2},...)\|^2&=&\sum_{k=1}^\infty|x_{n+k}|^2&\to& 0 \end{array}$$

Answer 2

Neither of 1, 2 implies the other. For a counterexample, consider the forward shift $$(x_1,x_2,x_3,\dots)\mapsto (0,x_1,x_2,\dots)$$ and the backward shift $$(x_1,x_2,x_3,\dots)\mapsto (x_2,x_3,x_4,\dots)$$ Each is the adjoint of the other one. The iterations of one of them converge to $0$, the iterations of the other do not.