Funny you should ask. My former student, Bentuo Zheng, and my visitor, Dongyang Chen, are in the process of developing this theory. The "right" definition involves the Pietsch factorization diagram, and is an off-shoot off what Farmer and I did for p-summing and p-integral operators (get the paper from my home page). Send me an email and I'll put you in contact with them.
On a related topic, my current student Alejandro ChavezDominguez is developing the operator ideal theory connected to non-linear p-summing operators and related mappings. This is something I have been interested in for a long time but did not see what to do (even the duality theory). In Javier's hands, the theory is developing very well.
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