L-weakly and M-weakly compact operators and the centre

Abstract

Abstract. We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide. Mathematics Subject Classification. 46B42, 47B60, 47B65.Keywords. L-weakly compact operator, M-weakly compact operator, Banach lattice, Centre of ordered vector space. 1.Introduction. An operator T on an ordered vector space E, with positive cone E + , is called central if it is bounded by a multiple of the identity operator, i.e., there exists some scalar α > 0 such that −αx ≤ T x ≤ αx holds for all x ∈ E + , i.e., −αI E ≤ T ≤ αI E , where I E denotes the identity operator on E. The collection of all central operators on E is called the centre of E and is denoted by Z(E). If E is Archimedean, then Z(E) is a commutative algebra and lattice with an order unit under composition, and is in fact isometrically algebra isomorphic to a dense subalgebra of some space, C(X), of continuous real-valued functions on some compact Hausdorff space X under pointwise multiplication and the supremum norm. Amongst many works on the centre, especially in the context of vector lattices, there have been attempts to identify the centre of various spaces of regular operators. In [9], isometric results were given by Wickstead showing that if E and F are Banach lattices, then there is an embedding of the algebraic tensor product Z(E) Z(F ) into Z L r (E, F ) which is an isometry when Z(E) Z(F ) is given the injective tensor product norm. That paper also gives some density results regarding this, and similar,