In 1970, V.D. Milman (Operators of class $C_{0}$ and $C^{*}_{0}$, Teor. Funkc. Funkc. Anal. Ih Priloz. 10(1970),15-26) proved that the product of two strictly singular operators on $L_{p}[0,1](1\leq p<\infty)$ or on $C[0,1]$ is compact. But, this fundamental paper is written in Russian and could not be downloaded. Who could give a detailed proof of this result? Thank you!
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asked Jun 19, 2016 at 14:49
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1$\begingroup$ Why don't you go to a library? (In any case you should give a reference to the paper.) $\endgroup$– Anton PetruninJun 19, 2016 at 22:02
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$\begingroup$ Anton, I give the reference to the paper. Sorry about it. $\endgroup$– Dongyang ChenJun 19, 2016 at 22:57
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$\begingroup$ I do not think that anyone will be willing to translate and compile the whole proof for you. You can try to look at the papers citing this paper (MathSciNet shows 20 such papers): maybe they contain generalizations. You can also check Milman's preliminary announcement which was published in Funkt. Anal. (vol. 3 (1969) no. 1, 93–94), and, I think, was translated into English. $\endgroup$– Mikhail OstrovskiiJun 20, 2016 at 13:44
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$\begingroup$ Thanks, Mikhail. I will try to look at the papers you mentioned. $\endgroup$– Dongyang ChenJun 20, 2016 at 14:22
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2$\begingroup$ Read page 2 of my paper "Multiplication operators..." with Schechtman, which you can download from my home page. There are enough hints there that you should be able to fill in the details and there are references to sources that contain more. This is for the $L_p$ case. The $C(K)$ space case can be found in books; I think Albiac-Kalton. The point is that an operator on $C(K)$ is strictly singular iff it is weakly compact, so the Dunford-Pettis property of $C(K)$ (i.e., that the image of a weakly compact set under a weakly compact operator is compact) gives the result. $\endgroup$– Bill JohnsonJun 20, 2016 at 15:57
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