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Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x \ne 0$ in X.

We have $||T^{-k}x||\leq M||x||$ for all $x\in X$. In particular, $||T^{-k}Tx||\leq M||Tx||$ i.e., $||T^{-k-1}x||\leq M||Tx||$.

Repeating these calculations we see that $||x||\leq M||T^nx||$ for all $x\in X$. So infimum of $||T^n x||$ is atleast $\frac{1}{M} ||x||\geq 0$ which is zero if and only if $x=0$.

So, $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x \ne 0$ in X.

I have not used that $X$ is Banach anywhere in the argument.

Is there any mistake in my argument?

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  • $\begingroup$ This seems correct. $\endgroup$ – Rick Sanchez Apr 23 '16 at 17:24
  • $\begingroup$ @RickSanchez : Then what about banach space $X$? $\endgroup$ – user311526 Apr 23 '16 at 18:59
  • $\begingroup$ I can't see where it's needed $\endgroup$ – Rick Sanchez Apr 23 '16 at 19:02
  • $\begingroup$ @RickSanchez : Can i delete the question then? $\endgroup$ – user311526 Apr 23 '16 at 19:03
  • $\begingroup$ There's no need to delete it, unless you want to. $\endgroup$ – Rick Sanchez Apr 23 '16 at 19:05

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