(p, q)-Compactness in spaces of holomorphic mappings

Abstract

Based on the concept of (p, q)-compact operator for p ∈ [1, ∞] and q ∈ [1, p*], we introduce and study the notion of (p, q)-compact holomorphic mapping between Banach spaces. We prove that the space formed by such mappings is a surjective pq∕(p + q)-Banach bounded-holomorphic ideal that can be generated by composition with the ideal of (p, q)-compact operators. In addition, we study Mujica’s linearization of such mappings, its relation with the (u*v* + tv* + tu*)∕tu*v*-Banach bounded-holomorphic composition ideal of the (t, u, v)-nuclear holomorphic mappings for t, u, v ∈ [1, ∞], its holomorphic transposition via the injective hull of the ideal of (p, q*, 1)-nuclear operators, the Möbius invariance of (p, q)-compact holomorphic map- pings on D, and its full compact factorization through a compact holomorphic mapping, a (p, q)-compact operator, and a compact operator.