Restriction of operators on $l_\infty$ to $c_0$

Question

Given $\epsilon>0$, can we always find a non-compact operator $T:l_\infty\to l_\infty$ of norm larger than $1$ such that the restriction of $T$ to $c_0$ is compact and has norm smaller than $\epsilon$?

Answer 1

There are two things that will help: First, whenever $c_0$ is isomorphic to a subspace of a separable Banach space $X$ then that subspace is complemented in $X$. Second, whenever $Y$ is a subspace of a normed space $X$ and $T:Y\to\ell_\infty$ is a continuous linear operator then there exists a continuous linear extension $\widetilde{T}:X\to\ell_\infty$ with $\|\widetilde{T}\|=\|T\|$.

Now we simply find a linear independent sequence $(y_n)_{n=1}^\infty\subseteq \ell_\infty\setminus c_0$. As $Z:=\overline{c_0+[y_n]_{n=1}^\infty}$ is a separable closed subspace of $\ell_\infty$ containing $c_0$, we can decompose it $Z=c_0\oplus Y$ for some infinite-dimensional subspace $Y$ of $\ell_\infty$. Write $I_{c_0}\in\mathcal{L}(c_0)$ andt $I_Y\in\mathcal{L}(Y)$ for the identity maps. Then for any $r\in\mathbb{R}$ we have $0I_{c_0}\oplus rI_Y\in\mathcal{L}(c_0\oplus Y)=\mathcal{L}(Z)$. Let $T$ denote its extension of $\ell_\infty$. Then $T|_{c_0}=0I_{c_0}$ is the zero mapping, trivially compact and with zero norm. On the other hand $\|T\|=r$, which we can make arbitrarily large.

Answer 2

One can even have $T|_{c_0}=0$. Indeed, first note that if $X\subseteq \ell_\infty$ is isomorphic to $\ell_2$, then $X\cap c_0$ is finite-dimensional so there is a subspace $X\subseteq \ell_\infty$ isometric to $\ell_2$ with $X\cap c_0=\{0\}$. (Actually, by simple repetition of the coordinates in the standard way of embedding separable spaces into $\ell_\infty$ one may guarantee from the start that the image has trivial intersection with $c_0$.) Secondly, $\ell_\infty / c_0$ has a quotient isomorphic to $\ell_2$ because it has (complemented) subspaces isomorphic to $\ell_\infty$ and the latter space has this property.

To finish the proof, take any linear surjection $Q\colon \ell_\infty/c_0\to \ell_2$ and let $S\colon \ell_2\to X$ be an isomorphic embedding. Let $\pi\colon \ell_\infty\to \ell_\infty / c_0$ be the quotient map. It is enough then to take $T=cSQ\pi$ for $c$ large enough.