Inequivalent Measures of Noncompactness and the Radius of the Essential Spectrum

The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, ‖ · ‖) assigns to each bounded set S in X a nonnegative real α(S) by the formula α(S) = inf{δ > 0 | S = n ⋃ i=1 Si for some Si with diam(Si) ≤ δ, for 1 ≤ i ≤ n <∞}. In general a map β which assigns to each bounded set S in X a nonnegative real and which shares most of the properties of α is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC’s β and γ on X are called equivalent if there exist positive constants b and c with bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets S ⊂ X. There are many results which prove the equivalence of various homogeneous MNC’s. Working with X = `p(N) where 1 ≤ p ≤ ∞, we give the first examples of homogeneous MNC’s which are not equivalent. Further, if X is any complex, infinite dimensional Banach space and L : X → X is a bounded linear map, one can define ρ(L) = sup{|λ| | λ ∈ ess(L)} where ess(L) denotes the essential spectrum of L. One can also define β(L) = inf{λ > 0 | β(LS) ≤ λβ(S) for every S ∈ B(X)}. The formula ρ(L) = lim m→∞ β(L) is known to be true if β is equivalent to α, the Kuratowski MNC; however, as we show, it is in general false for MNC’s which are not equivalent to α. On the other hand, if B denotes the unit ball in X and β is any MNC, we prove that ρ(L) = lim sup m→∞ β(LB) = inf{λ > 0 | lim m→∞ λ−mβ(LmB) = 0}. Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps.