Commutators on Banach Spaces



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A natural problem that arises in the study of derivations on a Banach algebra is to classify the commutators in the algebra. The problem as stated is too broad and we will only consider the algebra of operators acting on a given Banach space X. In particular, we will focus our attention to the spaces $\lambda I and \linf. The main results are that the commutators on 1 are the operators not of the form λI+K with λ≠0 and K compact and the operators on \linf which are commutators are those not of the form λI+S with λ≠0 and S strictly singular. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain these results 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 1≤p or p=0. In particular, it is shown that every non - E operator on L1 is a commutator. A characterization of the commutators on p1p2⊕⋯⊕pn is also given.