# Dosev, Detelin (2009-08). Commutators on Banach Spaces. Doctoral Dissertation. Thesis

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• Overview
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### abstract

• 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$. The main results are that the commutators on$\ell_1$are the operators not of the form$\lambda I + K$with$\lambda\neq 0$and$K$compact and the operators on$\linf$which are commutators are those not of the form$\lambda I + S$with$\lambda\neq 0$and$S$strictly singular. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain these results and use this generalization to obtain partial results about the commutators on spaces$\mathcal{X}$which can be represented as$\displaystyle \mathcal{X}\simeq \left ( \bigoplus_{i=0}^{\infty} \mathcal{X}\right)_{p}$for some$1\leq p\leq\infty$or$p=0$. In particular, it is shown that every non -$E$operator on$L_1$is a commutator. A characterization of the commutators on$\ell_{p_1}\oplus\ell_{p_2}\oplus\cdots\oplus\ell_{p_n}$is also given. • 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 $ell_1$ are the operators not of the form $lambda I + K$ with $lambda eq 0$ and $K$ compact and the operators on $linf$ which are commutators are those not of the form $lambda I + S$ with $lambda eq 0$ 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
$mathcal{X}$ which can be represented as $displaystyle mathcal{X}simeq left ( igoplus_{i=0}^{infty} mathcal{X} ight)_{p}$ for some $1leq pleqinfty$ or $p=0$.
In particular, it is shown that every non - $E$ operator on $L_1$ is a commutator. A characterization of the commutators on $ell_{p_1}oplusell_{p_2}opluscdotsoplusell_{p_n}$ is also given.

• August 2009