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Approximation properties determined by operator ideals and approximability of homogeneous polynomials and holomorphic functions

Sonia Berrios;  Geraldo Botelho — 2012

Studia Mathematica

Given an operator ideal ℐ, a Banach space E has the ℐ-approximation property if the identity operator on E can be uniformly approximated on compact subsets of E by operators belonging to ℐ. In this paper the ℐ-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of linear operators/homogeneous polynomials, spaces of holomorphic functions and their preduals.

When is the Haar measure a Pietsch measure for nonlinear mappings?

Geraldo Botelho;  Daniel Pellegrino;  Pilar Rueda;  Joedson Santos;  Juan Benigno Seoane-Sepúlveda — 2012

Studia Mathematica

We show that, as in the linear case, the normalized Haar measure on a compact topological group G is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of C(G). This answers a question posed to the authors by J. Diestel. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section some problems are proposed.

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