The domination theorem for operator classes generated by Orlicz spaces

We study lattice summing operators between Banach spaces focusing on two classes, ℓφ$\ell _\varphi$‐summing and strongly φ$\varphi$‐summing operators, which are generated by Orlicz sequence lattices ℓφ$\ell _\varphi$. For the class of strongly φ$\varphi$‐summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for p$p$‐summing operators. Based on this result, we show that strongly φ$\varphi$‐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 φ$\varphi$‐summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when ℓφ$\ell _\varphi$ is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that ℓφ$\ell _\varphi$‐nuclear operators are ℓφ$\ell _\varphi$‐summing when ℓφ$\ell _\varphi$ is separable.