Multiplication operators on L(Lp) and lp-strictly singular operators Academic Article uri icon

abstract

  • A classification of weakly compact multiplication operators on L(L p), 1 < p < ∞, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of l p-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, 1 < p < 2, is l p-strictly singular and T|x is an isomorphism for some subspace X of Lp, then X embeds into Lr for all r < 2, 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 ≤ p < 2, and T|x is an isomorphism for some reflexive subspace X of L p, then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976. © European Mathematical Society 2008.

author list (cited authors)

  • Johnson, W., & Schechtman, G.

complete list of authors

  • Johnson, William||Schechtman, Gideon

publication date

  • January 1, 2008 11:11 AM