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Given a Banach space $X$ and a bounded linear map $T:X\rightarrow X$ we define $$e^T = I + \sum_{n\geq1}\frac{T^n}{n!}$$ Show that if $e^T$ is compact then dim $X<\infty$.

I have showed before that $e^T$ converges in operator norm and obviously it commutes with $T$. Hints are welcome!

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Hint: for which $T$ is $e^T$ invertible? (Think about the case $X=\mathbb{R}$).

Invertibility of compact operators in infinite-dimensional Banach spaces might help.

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