Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\lVert T\rVert_e=\lVert T\rVert$, is it true that (or when is it true that) $\lVert T^{-1}\rVert=\lVert T^{-1}\rVert_e$? Here, $\lVert\cdot\rVert_e$ denotes the essential norm.
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4$\begingroup$ It is certainly not true in general. For example, write $X$ as the direct sum of an infinite dimensional closed subspace $Y$ and a one dimensional subspace $Z$ and consider an operator that is the identity on $Y$ and a small positive multiple of the identity on $Z$, so that in the Calkin algebra the operator is the identity element. $\endgroup$– Bill JohnsonCommented Aug 25, 2017 at 19:59
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