Generating Ideals of Bloch Mappings via Pietsch’s Quotients

In this paper, we introduce the notion of the normalized Bloch left-hand quotient ideal ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$, where $\mathcal{A}$ is an operator ideal and ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a normalized Bloch ideal, as a nonlinear extension of the concept of the left-hand quotient of operator ideals. We show that these quotients constitute a new method for generating normalized Bloch ideals, complementing the existing methods of generation by composition and transposition. In fact, if ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the linearization property in a linear operator ideal $\mathcal{J}$, then ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a composition ideal of the form $({\mathcal{A}}^{-1}\circ \mathcal{J})\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$. We conclude this work by introducing two important subclasses of Bloch maps; these are Bloch maps with the Grothendieck and Rosenthal range. We focus on showing that they form normalized Bloch ideals which can be seen as normalized Bloch left-hand quotients ideals. In addition, we pose an open problem concerning when a Bloch quotient without the linearization property in an operator ideal cannot be related to a normalized Bloch ideal of the composition type, for which we will use the subclass of p-summing Bloch maps.