# Example of operator with $\lVert T \rVert^ i = 1$ in normed spaces

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.

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