On isometric isomorphisms of the Bloch space on the unit ball of \({\mathbb{C}}^ n\) (Q1262981)

The isometric isomorphisms of the closed subspace S, spanned by polynomials, of the Bloch space on the unit ball \(B^ n\) in \({\mathbb{C}}^ n\) is characterized. In fact such an isometric isomorphism is given, up to suitable normalizations, by composition with a Möbius transformation. Theorem. Let U: \(S\to S\) be an isometric isomorphism. Then there is a \(\phi \in Aut(B^ n)\) and a \(\mu\in {\mathbb{C}}\) \((| \phi | =1)\) with \(Uf(z)=\mu (f(\phi (z))-f(\phi (0)))\) for every \(f\in S.\) A similar result in the strongly pseudoconex case is conjectured to be true by the authors.