A core band of a bounded operator in the order-continuous ideal of a Banach lattice

Every bounded operator T : E Y from a Banach lattice E into a Banach space Y admits a canonical localisation on the order-continuous ideal E a .A band BT E a is introduced as the largest band of E a on which T is AM-compact.Equivalently, an element x E a belongs to BT precisely when T ([-|x|, |x|]) is relatively compact in Y .This order-interval compactness is quantified by a lattice seminorm T on E a , defined via the Kuratowski measure of noncompactness of T ([-|x|, |x|]).The kernel of T coincides with BT , thereby establishing a link with the restriction characterisation for LW-compact operators.