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‎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 ‎‎$‎T‎$‎.)

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    No, consider for example $T=\frac12\mathbb 1$. – 2017-01-17
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    Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see [Guidelines for good use of $\LaTeX$ in question titles](http://meta.math.stackexchange.com/a/9730). – 2017-01-17
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    $\sigma(T) \subseteq \{ \lambda\in\mathbb{C} : |\lambda| \le 1 \}$. – 2017-01-17
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    ‎ how can we prove? $ ‎\sigma (‎ T‎ )‎ ‎‎\subseteq‎ ‎\{‎ ‎\lambda ‎\in ‎‎\mathbb{C} : ‎‎\parallel ‎\lambda ‎\parallel ‎\leq 1 \}‎‎‎‎‎‎$‎ – 2017-01-22
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    It is a different story if $\{\|T^n\|: n\in\mathbb Z\}$ is bounded. – 2017-04-13

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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 $\sigma(T)=\{1,\frac{1}{2}\}$.