ON ABSOLUTELY SUMMING OPERATORS ON C(K) WITH VALUES IN BANACH LATTICES

Lindenstrauss and Pełczyński¹³ in 1968 proved that for a compact Hausdorff space K every bounded linear operator on C(K) with values in some L_p(μ), 2 ≤ p < ∞ is absolutely (p, 2)-summing. More generally, Maurey¹⁷ in 1973/74 showed that this is also valid for any Banach space X of cotype p instead of L_p(μ), and Defant and the author⁶ in 2000 proved the case where X is a p/2-Hilbert space instead of ℓ_p. In this note, we investigate refinements of the above results within the more general setting of absolutely (E, 2)-summing operators and uniformly convex Banach lattices. We apply our results to obtain lower estimate for Gelfand numbers of certain finite-dimensional identity maps.