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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2009-08-6994 |
| Date | 2009 August 1900 |
| Creators | Dosev, Detelin |
| Contributors | Johnson, William B. |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | application/pdf |
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