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Can one give me an easy example of an operator $T$ on a Banach space which is injective and has closed range and such that $\|T^2\|\neq \|T\|^2$?

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up vote 7 down vote accepted

Consider $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ as an operator on $\mathbb C^2$ with standard inner product.

(Note that in case the Banach space is a Hilbert space and $T$ is normal, the identity $\|T^2\|=\|T\|^2$ always does hold, so I just wrote down the first nonnormal invertible matrix I thought of, and it works.)

Added: Here's a rough way to see why matrices of the form $T=\begin{bmatrix}1&a\\0&1\end{bmatrix}$ will do the trick. When $|a|$ is very large, $\|T\|\approx |a|$, and $\|T^2\|\approx 2|a|$, while $\|T\|^2\approx |a|^2$ is much larger than $2|a|$.

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+1 for explaining the rationale of "try a pathological case to see if it's a counterexample". – Hurkyl Dec 17 '12 at 19:45

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