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

Abstract

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

Keywords

• vector-valued holomorphic function
• linearization
• factorization theorems
• (p, q)-compact operator
• (p, q)-compact holomorphic mapping
• (t, u, v)-nuclear holomorphic mapping
• 47b07
• 46e15
• 46e40
• 47l20