On (p, r, s)-summing Bloch maps and Lapresté norms

Abstract

The theory of (p, r, s)-summing and (p, r, s)-nuclear linear operators on Banach spaces was developed by Pietsch in his book on operator ideals (Pietsch in Operator ideals, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980, Chapters 17 and 18) Due to recent advances in the theory of ideals of Bloch maps, we extend these concepts to Bloch maps from the complex open unit disc \(\mathbb {D}\) into a complex Banach space X. Variants for (r, s)-dominated Bloch maps of classical Pietsch’s domination and Kwapień’s factorization theorems of (r, s)-dominated linear operators are presented. We define analogues of Lapresté’s tensor norms on the space of X-valued Bloch molecules on \(\mathbb {D}\) to address the duality of the spaces of \((p^*,r,s)\)-summing Bloch maps from \(\mathbb {D}\) into \(X^*\). The class of (p, r, s)-nuclear Bloch maps is introduced and analysed to give examples of (p, r, s)-summing Bloch maps.