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Let X be a normed vector space of your choice with its norm $\lVert.\rVert$. I am looking for an operator of norm $\lVert T \rVert ^ i = 1$.

Defined on as $T:X \rightarrow X$ s.t. its powers $T^{i} = T \circ T^{i-1}$ for all $i \geq 1$.

Here is where I am having trouble though, it needs to have $T^{i} \neq T^{i-1}$

Any thoughts would be much appreciated.

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    $\begingroup$ Surely you mean $\lVert T^i\rVert = 1$, not $\lVert T\rVert^i = 1$? Anyway, just take an irrational rotation on $X = \mathbb C$. $\endgroup$ – LSpice May 16 at 19:56
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    $\begingroup$ Shift operator $\endgroup$ – A.Γ. May 16 at 20:14
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    $\begingroup$ As written, the condition $\|T\|^i = 1$ (for some/all $i$) is equivalent to $\|T\| = 1$, as $\|T\| \ge 0$. $\endgroup$ – Theo Bendit May 16 at 20:26
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    $\begingroup$ You can take $T=-Id$. $\endgroup$ – Severin Schraven May 16 at 21:52
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    $\begingroup$ @SeverinSchraven You came up with the simlpest example! $\endgroup$ – Kavi Rama Murthy May 16 at 23:49

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