The domination theorem for operator classes generated by Orlicz spaces

Abstract We study lattice summing operators between Banach spaces focusing on two classes, ‐summing and strongly ‐summing operators, which are generated by Orlicz sequence lattices . For the class of strongly ‐summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for ‐summing operators. Based on this result, we show that strongly ‐summing operators are Dunford–Pettis. As a consequence, we show that these classes are, in general, distinct. We also demonstrate that the class of strongly ‐summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that ‐nuclear operators are ‐summing when is separable.