On Pietsch measures for summing operators and dominated polynomials

TL;DR

This paper explores the relationship between Pietsch measures, summing operators, and dominated polynomials in Banach space theory, providing new insights into their injectivity and factorization properties.

Contribution

It establishes a connection between the injectivity of certain maps and the existence of injective p-summing operators or polynomials with specific Pietsch measures, filling gaps in existing proofs.

Findings

Characterizes injectivity of canonical maps in terms of Pietsch measures.

Links injective p-summing operators to measure-theoretic properties.

Completes proofs of Pietsch-type factorization results for dominated polynomials.

Abstract

We relate the injectivity of the canonical map from $C(B_{E'})$ to $L_p(\mu)$, where $\mu$ is a regular Borel probability measure on the closed unit ball $B_{E'}$ of the dual $E'$ of a Banach space $E$ endowed with the weak* topology, to the existence of injective $p$-summing linear operators/$p$-dominated homogeneous polynomials defined on $E$ having $\mu$ as a Pietsch measure. As an application we fill the gap in the proofs of some results of concerning Pietsch-type factorization of dominated polynomials.