Absolutely summing polynomials on Banach spaces with unconditional basis ✩

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define μX,(xn) = inf{t ; (aj ) ∈ t whenever x = ∞j =1 aj xj ∈ X} and obtain estimates for μX,(xn) 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. © 2005 Elsevier Inc. All rights reserved