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It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators.

Besides this, the results by Per Enflo 1973 shows that this results is not true in general on Banach spaces.

My question is to know whether, this results is true at least on $L^p$-spaces? That is

An operator a compact $T:L^p\to L^p$ is limit of finite rank operators?

I am looking for references or any results in this direction.

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    $\begingroup$ Yes, because these spaces have a basis. (For the case $L_\infty$ you can use a different argument). The approximation property is the term to look up. $\endgroup$
    – Tomasz Kania
    Commented Jan 31, 2022 at 11:16
  • $\begingroup$ You mean every Banach space with a Basis has this property? Especially separable spaces? $\endgroup$
    – Guy Fsone
    Commented Jan 31, 2022 at 19:45
  • $\begingroup$ math.stackexchange.com/questions/923634/… $\endgroup$
    – Tomasz Kania
    Commented Jan 31, 2022 at 20:11

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