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

A well-known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existenceof a universal non-weakly compact operator between Banach spaces. We show theexistence of universal non-Rosenthal, non-limited, and non-Grothendieck operators.We also prove that there does not exist a universal non-Dunford–Pettis operator, butthere 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