Absolutely summing polynomials on Banach spaces with unconditional basis

Abstract

If X is a Banach space with a normalized unconditional Schauder basis ( x n ) , we define μ X , ( x n ) = inf { t ; ( a j ) ∈ ℓ t whenever x = ∑ j = 1 ∞ a j x j ∈ X } and obtain estimates for μ X , ( x n ) when every continuous m-homogeneous polynomial from X into Y is absolutely ( q , 1 ) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m = 1 , we obtain new contributions to the linear theory of absolutely summing operators.