# Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non-nuclear operator is the norm limit of finite rank operators?

• Isn't this equivalent to asking: is there a space X on which every approximable operator is nuclear? (and so you want to assume X infinite-dimensional to avoid trivialities) – Yemon Choi May 1 '16 at 15:11
• I believe it is an open question if there exists a Banach space X for which every operator is "scalar plus nuclear". If such a space existed it would have the property that every compact operator is nuclear, which seems to satisfy your requirements – Yemon Choi May 1 '16 at 15:13
• You are right, that would do it, but I think my question is weaker. In "scalar plus nuclear" there would not be any compact, non-nuclear operators. – Markus May 1 '16 at 15:16
• Is the folloing space appropriate? – Ali Taghavi May 2 '16 at 19:27
• There is a banach space such that the only bounded operators are the compact perturbation of scalar operators. – Ali Taghavi May 2 '16 at 19:30

• I don't know if Pisier's construction always gives a space on which there operators that are not scalar plus nuclear, but they can contain $\ell_1\oplus \ell_2$ isomorphically, and if follows from known results that on any space that contains $\ell_1\oplus \ell_2$ isomorphically there is an operator that has closed, infinite dimensional and infinite codimensional.range. – Bill Johnson May 2 '16 at 23:14