(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 p q ∕ ( 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 * + t v * + t u * ) ∕ t u * 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.