A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds $$ \mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N})) $$ Now let $S_1,S_2$ two infinite sets and at least one is uncountable. Is it still true that $$ \mathcal{B}(\ell_r(S_1),\ell_p(S_2))=\mathcal{K}(\ell_r(S_1),\ell_p(S_2)\quad ? $$ I've tried to mimic standard proof of Pitt's from Albiac Kalton, but got stuck since unit ball of $\ell_r(S_1)$ with weak topology not necesseary metriazable.
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Yes.
Rosenthal, H. P. On quasi complemented subspaces of Banach spaces with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\nu)$. J. Functional Analysis 4, 176--214 (1969), Theorem A2, page 206: http://dx.doi.org/10.1016/0022-1236(69)90011-1
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H.E. Lacey, The isometric theory of Classical Banach spaces, Theorem 13, page 129.
