In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc.
Given an operator ideal I, we study the multi-ideal I • L and the polynomial ideal I • P. The connection with the linearizations of these mappings on projective symmetric tensor products is investigated in detail. Applications to the ideals of strictly singular and absolutely summing linear operators are obtained. §1. Introduction Since the 1983 paper by A. Pietsch [26], ideals of multilinear mappings (multi-ideals) and homogeneous polynomials (polynomial ideals) between Banach spaces have been studied as a natural consequence of the successful theory of operator ideals. Several ideals have been investigated and abstract methods to generate ideals of multilinear mappings and polynomials have been introduced (see [6,20]
We prove that, like in the linear case, there is a canonical prototype of a p-dominated homogeneous polynomial through which every p-dominated polynomial between Banach spaces factors.
Given a surjective ideal of operators, we undertake a new general procedure to construct an ideal of polynomials. The relation with the ideal of polynomials obtained by the well-known method of composition is established.
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