Measures of Non-compactness of Operators on Banach Lattices

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[Indag. Math.(N.S.)2(2) (1991), 149–158; Uspehi Mat. Nauk27(1(163)) (1972), 81–146] used representation spaces to study measures of non-compactness and spectral radii of operators on Banach lattices. In this paper, we develop representation spaces based on the nonstandard hull construction (which is equivalent to the ultrapower construction). As a particular application, we present a simple proof and some extensions of the main result of [J. Funct. Anal. 78(1) (1988), 31–55] on the monotonicity of the measure of non-compactness and the spectral radius of AM-compact operators. We also use the representation spaces to characterize d-convergence and discuss the relationship between d-convergence and the measures of non-compactness.

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