Diagonals of the Powers of an Operator on a Banach Lattice

This paper is devoted to a detailed study of the properties of the band projection D of the complete lattice ordered algebra −_r(E) of the regular (or order bounded) operators of a Dedekind complete Banach lattice E onto the center Z(E) of E. We recall that the center Z(E) is the commutative subalgebra of −_r(E) of all T satisfying |T| ≤ λI, where I is the identity operator. In the finite dimensional case, with respect to the standard numerical basis, Z(E) is the algebra of all diagonal matrices. For this reason the band projection D is called the diagonal map of E.