On quotients of ideals of bounded holomorphic maps

Abstract

Based on the notion of left-hand quotient of operator ideals, we introduce and study the concept of bounded-holomorphic left-hand quotient $\I^{-1}\circ\J^{\H^\infty}$, where $\I$ is an operator ideal and $\J^{\H^\infty}$ is a bounded-holomorphic ideal. We show that such quotients are a method for generating new bounded-holomorphic ideals. In fact, if $\J^{\H^\infty}$ has thelinearization property in an operator ideal $\A$, then$\I^{-1}\circ\J^{\H^\infty}$ is a composition ideal of the form $(\I^{-1} \circ \A) \circ \H^\infty$. We also introduce the notion of Grothendieck holomorphic map and prove that they form a bounded-holomorphic ideal which can be seen as a bounded-holomorphic left-hand quotient. In the same way, the ideal of holomorphic maps with Rosenthal range can be generated as a bounded-holomorphic left-hand quotient.