Let $X,Y$ be Banach spaces. We denote by $\mathcal{K}(X,Y)$ the space of all compact operators from $X$ into $Y$. For an operator $T:X\rightarrow Y$, we let $$\|T\|_{e}:=\inf\{\|T-K\|:K\in \mathcal{K}(X,Y)\},$$ and $$\|T\|_{m}:=\inf\{\|T|_{M}\|:codim M<\infty\},$$ where $M$ represents the finite co-dimensional subspace of $X$. It is known that $T$ is compact if and only if $\|T\|_{m}=0$.

Let $X$ be a Banach space. We define $J:X\rightarrow l_{\infty}(B_{X^{*}})$ by $\langle Jx,x^{*}\rangle=\langle x^{*},x\rangle, x\in X, x^{*}\in B_{X^{*}}.$ Let $T:Z\rightarrow X$ be an operator. My question is:

Question. Does $\|JT\|_{e}=\|T\|_{m}$ hold?

History of Banach Spaces and Linear OperatorsandSpectral Theory and Differential Operatorsby Edmunds and Evans. Both discuss s-numbers at length. If I remember right $\Vert T \Vert_m$ looks like it might be connected with the so-called `Gelfand numbers' of $T$ $\endgroup$ – DCM Apr 12 '20 at 14:45