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.



Pitt's theorem for non-separable $\ell_p$ spaces
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>


or 


H.E. Lacey, The isometric theory of Classical Banach spaces, Theorem 13, page 129.


