Banach space duality of absolute Schur algebras

Abstract

Matrix Schur product is the entry-wise product of matrices of the same size. It was shown by P. Chaisuriya and S.-C. Ong [1] that (forr≥1) infinite matrices [ajk] such that [|ajk|r] ɛB(l2 form a Banach algebra under the norm ‖[ajk]‖r=‖[|ajk|r]‖1/r and the Schur product. In this paper we demonstrate the existence of Banach space duality within the class of these algebras which is analogous to the classical duality between the spaces of compact, trace class, and bounded operators onl2. Also we obtain a general functional calculus on these algebras, which is used to determine the spectrum and to justify the notion of ∞-norm introduced in [1].