arXiv subjects
Publications and source records attributed to Detelin Dosev.
The operators on $\LP=L_p[0,1]$, $1\leq p<\infty$, which are not commutators are those of the form $λI + S$ where $λ\neq 0$ and $S$ belongs to the largest ideal in $\opLP$. The proof involves new structural results for operators on $\LP$ which are of independent interest.
The operators on $\ell_{\infty}$ which are commutators are those not of the form $λI + S$ with $λ\neq 0$ and $S$ strictly singular.
The main result is that the commutators on $\ell_1$ are the operators not of the form $λI + K$ with $λ\neq 0$ and $K$ compact. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 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 $\displaystyle \X\simeq (\bigoplus_{i=0}^{\infty} \X)_{p}$ for some $1\leq p<\infty$ or $p=0$. In particular, it is shown that every compact operator on $L_1$ is a commutator. A characterization of the commutators on $\ell_{p_1}\oplus\ell_{p_2}\oplus...\oplus\ell_{p_n}$ is given. We also show that strictly singular operators on $\ell_{\infty}$ are commutators.