# Extending compact operators into $c_0$

Lindenstrauss has the following paper: http://www.ams.org/journals/bull/1962-68-05/S0002-9904-1962-10787-3/S0002-9904-1962-10787-3.pdf

I would like to see the proof for the following theorem (from the above paper):

Theorem: Let $X,Y,Z$ be Banach spaces with $Y\subset Z$. Suppose $X^*=L_1(\mu)$ for some measure $\mu$. Then every compact operator $T:Y\to X$ has, for every $\epsilon>0$, a compact extension $\widetilde{T}:Z\to X$ with $\|\widetilde{T}\|\leq(1+\epsilon)\|T\|$.

Unfortunately, Lindenstrauss's proof for this theorem consists of the line "due to Grothendieck." He then gives a reference to a paper by Grothendieck, in French.

Grothendieck's paper is here (in French): http://matematicas.unex.es/~navarro/res/ega/Carat%C3%A9risation%20des%20Espaces%20L1.pdf

Unfortunately I only know a little French, and even with Google translate to help, I couldn't even decipher the theorem statements, much less their proofs!

So my question is this: Does anyone know a textbook or a paper in English which contains a detailed proof to the above theorem?

Thanks!

EDIT: By the way, I might as well give some background too. Let $X=c_0(\mathbb{N})$. I am given an operator $T:X\to X$ which satisfies some very funky conditions including (but not limited to) $\sigma_p(T^*)=\emptyset$ and $\partial\sigma(T)\subseteq\sigma_p(T)$, and $\sigma(T)$ uncountable. The set of all $T$-eigenvectors of norm $\leq 1$ is compact. I don't want to list all the conditions due to space, but those are the most striking. I am able to show that there is a closed subspace $Y\subset X$ which is both infinite-dimensional and infinite-codimensional in $X$, and for which the restriction $T|_Y:Y\to X$ is compact. My ultimate goal is to find an operator $S:X\to X$ with countable point spectrum which commutes with $T$ and which is not a multiple of the identity. So if $S$ is compact and commutes with $T$, that will do it. The above theorem allows us to find a compact extension of $T|_Y$ to all of $X$. I hope that this extension is such that it commutes with $T$. Alternatively, perhaps it can give me ideas as to how to construct such an operator.

• Did you look at Lindensrauss' Memoirs? – Bill Johnson Aug 11 '13 at 22:54

If you don't care about the norm of an extension of your compact operator, you can give a short proof of this theorem using the fact that the operation of taking the projective tensor product with $L_1(\mu)\cong C(K)^*$ respects quotients.
Regarding your motivations, once you have your subspace $Y$, why can't you pass to a further subspace $Y_0\subset Y$ still having infinite codimension but additionally isomorphic to $c_0$ hence complemented? ($c_0$ is saturated by subspaces isomorphic to $c_0$.) Then you can extend $T|_{Y_0}$ to the whole space by 0.
• Awesome, thank you for the reference! Regarding your comment, it's a good idea and one that I had considered. However the problem is commutation. Suppose $X=Y_0\oplus Z_0$ where $S:=T|_{Y_0}\oplus 0$ is compact. We need to show that $TS=ST$. Then given $y+z\in X$ we have $TS(y+z)=T^2y$, but $ST(y+z)=T^2y$ only if $Tz\in Y_0$. Still, it's a good idea and one I intend to keep in mind as I research. – Ben W Aug 12 '13 at 13:36