If $ T \in B ( H ) $ is invertible and for all $ n \geq 1 $, $ \parallel T^{n} \parallel $is bounded, The following statement is true?
$ \sigma ( T ) \subset \{ \lambda \in \mathbb{C} : \mid \lambda \mid = 1 \}$
($ \sigma ( T ) $ is spectrum of $T$.)
The above statement is not true. Let $H= \mathbb C^2$ and $T$ given by
$$T(z_1,z_2)=(z_1, \frac{1}{2}z_2).$$
We have $||T^n||=1$ for all $n$ and $\sigma(T)=\{1,\frac{1}{2}\}$.