Multiplication operators on L(Lp) and ℓp-strictly singular operators

• Main Author: William B. Johnson
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 2007
• Subjects: Rosenthal; tylli
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.78.3974; http://www.wisdom.weizmann.ac.il/~gideon/papers/JSElemOp.pdf

A classification of weakly compact multiplication operators on L(Lp), 1 < p < ∞, 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, 1 < p < 2, is ℓp-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 Lp, then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.