Commutators on 1

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Abstract

The main result is that the commutators on 1 are the operators not of the form λI+K with λ0 and K compact. We generalize Apostol's technique [C. Apostol, Rev. Roumaine Math. Appl. 17 (1972) 1513–1534] to obtain this result and use this generalization to obtain partial results about the commutators on spaces X which can be represented as X(i=0X)p for some 1p< or p=0. In particular, it is shown that every compact operator on L1 is a commutator. A characterization of the commutators on p1p2pn is given. We also show that strictly singular operators on are commutators.

Keywords

Commutators

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1

Research supported in part by NSF grant DMS-0503688.

2

This is part of the author's doctoral dissertation, which is being prepared at Texas A&M University under the direction of W.B. Johnson.