Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces

Abstract

Using the theory of full and symmetric tensor norms on normed spaces, a theorem of Kürsten and Heinrich on ultrastability and maximality of normed operator ideals is extended to ideals of n n -homogeneous polynomials and n n -linear mappings—scalar-valued and vector-valued. The motivation for these results is the following important special case: the “uniterated” Aron-Berner extension q ¯ U \overline {q}^{\mathfrak U} : E ⟶ F E \longrightarrow F of an n n -homogeneous polynomial q : E ⟶ F q: E\longrightarrow F to the bidual remains in certain ideals under preservation of the norm. Moreover, Lotz’s characterization of maximal normed ideals of linear mappings through appropriate tensor norms is proved for ideals of n n -homogeneous scalar-valued polynomials and ideals of n n -linear mappings.