Measures of non-compactness of operators in Banach lattices

Abstract

Let E and F be complex Banach lattices. Then a measure of non-semicompactness ϱ( T) is introduced for an order bounded operator T from E into F. If E ∗ and F have order continuous norm then for every AM-compact operator ϱ( T) = β( T), where β( T) denotes the ball measure of non-compactness of T. From this result monotonicity properties of β( T) and the essential spectral radius r ess ( T) are derived for AM-compact operators. Also shown is that r ess ( T) ϵ σ ess ( T) for positive AM-compact operators. In addition properties of the essential spectrum of norm bounded disjointness preserving operators are proved.