Geodesic
Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators
Journal of the European Mathematical Society, Tome 10 (2008) no. 4, pp. 1105-1119.

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A classification of weakly compact multiplication operators on L(Lp​), 1∞, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of lp​-strictly singular operators, and we also investigate the structure of general lp​-strictly singular operators on Lp​. The main result is that if an operator T on Lp​, 12, is lp​-strictly singular and T∣X​ is an isomorphism for some subspace X of Lp​, then X embeds into Lr​ for all r2, but X need not be isomorphic to a Hilbert space. It is also shown that if T is convolution by a biased coin on Lp​ of the Cantor group, 1≤p2, and T∣X​ is an isomorphism for some reflexive subspace X of Lp​, then X is isomorphic to a Hilbert space. The case p=1 answers a question asked by Rosenthal in 1976.
DOI : 10.4171/jems/141
Classification : 46-XX, 00-XX
Keywords: Elementary operators, multiplication operators, strictly singular operators, Lp​ spaces, biased coin 
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William B. Johnson; Gideon Schechtman. Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators. Journal of the European Mathematical Society, Tome 10 (2008) no. 4, pp. 1105-1119. doi : 10.4171/jems/141. http://geodesic.mathdoc.fr/articles/10.4171/jems/141/

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