| Benson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibility of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to the complex method of interpolation and to the real method of interpolation for the case q = infinity. Two new characterizations are presented of the domain space of the "derivation" operator O in the context of orbital methods. Comparisons to the work of others are made, especially the unifying paper of M. Cwikel, N. Kalton, M. Milman and R. Rochberg, A Unified Theory of Commutator Estimates for a Class of Interpolation Methods, Advances in Mathematics 169 (2002), 241-312. |
