Inequivalent measures of noncompactness and the radius of the essential spectrum

The Kuratowski measure of noncompactness a on an infinite dimensional Banach space (X,||.||) assigns to each bounded set S in X a nonnegative real number α(S) by the formula α(S) =inf{δ > 0 | S = U ⁿ _i=1S_i for some S_i with diam(S _i) ≥ δ, for 1 ≤ i ≤ n < ∞}. In general a map β which assigns to each bounded set S in X a nonnegative real number and which shares most of the properties of a 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 C X. There are many results which prove the equivalence of various homogeneous MNC's. Working with X = ℓ^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 ?(L) = sup{|γ| | γ ε ess(L)}, where ess(L) denotes the essential spectrum of L. One can also define β(L) = inf{γ > 0 | β(LS) ≤ γβ(S) for every S ε B(X)}. The formula ρ(L) = lim m→∞ β(L^m)1/m is known to be true if β is equivalent to a, 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 homogeneous MNC, we prove that ρ(L) = lim sup m→8 β(L^mB)^1/m = inf{γ > 0 | lim m→∞ γ^-mβ(L^mB) = 0}. Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps.