Inequivalent measures of noncompactness

Abstract

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called "equivalent" if there exist positive constants b and c such that bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets S ⊂ X. If such constants do not exist, the measures of noncompactness are "inequivalent." We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = Lp(Ω, Σ, μ); or if K is a compact Hausdorff space and X=C(K); or if K is a compact metric space, 0 < λ ≤ 1, and X=C0,λ(K), the Banach space of Hölder continuous functions with Hölder exponent λ. We also prove the existence of such inequivalent measures of noncompactness if Ω is an open subset of ℝn and X is the Sobolev space Wm,p(Ω). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f: C→C and from the closely related issue of giving the proper definition of the "cone essential spectral radius" of such maps. These questions are considered in the companion paper [28]; see, also,[27]. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.