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This paper has a twofold purpose: to present an overview of the theory of absolutely  summing operators and its different generalizations for the multilinear setting, and to sketch the beginning of a research project related to an objective search of “perfect” multilinear extensions of the ideal of absolutely summing operators. The final sections contain some open problems that may indicate lines for future investigation.

Keywords: Absolutely summing operators, multiple summing multilinear operators, strongly multiple summing multilinear operators, multi-ideals

Quaestiones Mathematicae 34(2011), 447–478