Banach space duality of absolute Schur algebras

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 (for r greater than or equal to 1) infinite matrices [a(jk)] such that [\a(jk)\(r)] E is an ele ment of B (l(2)) form a Banach algebra under the norm parallel to [a(jk)]pa rallel to (r) = parallel to[\a(jk)\(r)]parallel to (1/r) and the Schur prod uct. In this paper we demonstrate the existence of Banach space duality wit hin the class of these algebras which is analogous to the classical duality between the spaces of compact, trace class, and bounded operators on l(2). Also we obtain a general functional calculus on these algebras, which is u sed to determine the spectrum and to justify the notion of oc-norm introduc ed in [1].