In my functional analysis class I was asked this question which got me stuck:
Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying:
$ || T || < 1 $ in the operator norm. We are asked if the following necessarily is true: for all $ h \in \mathbb{H} $ does one have the convergence to zero $ T^nh \to 0 $ in the Hilbert space norm?
I cannot really think of any counterexamples or how to prove it is true so I need help and am asking here. Thanks to all helpers.