The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $\Omega \in L{\left(logL\right)}^{k+1}\left(S^{n-1}\right)$ is a sufficient condition for the kth order commutator to be bounded on $L^p\left(ℝⁿ\right)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

It is proved that if $J_i$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_i=N_i+Q_i$, where $N_i$ is normal and $Q_i$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.

It is proved that every locally inner derivation on a symmetric norm ideal of operators is an inner derivation.