Universal mappings for certain classes of operators and polynomials between Banach spaces

Abstract

A well-known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non-weakly compact operator between Banach spaces. We show the existence of universal non-Rosenthal, non-limited, and non-Grothendieck operators.We also prove that there does not exist a universal non-Dunford-Pettis operator, but there is a universal class of non-Dunford-Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non-weakly compact polynomials, there does not exist a universal polynomial. K E Y W O R D SIdeals of homogeneous polynomials, surjective operator ideals, universal operator, universal polynomial M S C ( 2 0 1 0 ) Johnson [27] showed in 1971 that the formal identity operator 1 → ∞ is universal for the class of non-compact operators. These results are useful in order to prove that a given operator is non-weakly compact (respectively, non-compact). In 1997, M. Girardi and W. B. Johnson [21] proved that there does not exist a universal non-completely continuous operator, but there is a class  of universal non-completely continuous operators, that is, for every non-completely continuous operator , there is some member of  that factors through .Here we prove the existence of universal operators for the classes of non-Rosenthal, non-Grothendieck, and non-limited operators and the existence of a universal class of non-Dunford-Pettis operators (see below the definitions of all such classes