Isometries on the vector valued little Bloch space
Let \(E\) be a smooth, strictly convex, reflexive complex Banach space. Let \(\Delta\) be the open unit disc. Let \({\mathcal B}(\Delta,E)\) and \({\mathcal B}_0(\Delta,E)\) be the Bloch and little Bloch spaces. Starting with a description of the extreme points of the dual unit ball, the authors obtain a complete description of the surjective isometries of the vector-valued function spaces, in terms of surjective isometries of the underlying Banach spaces, and an automorphism \(\sigma\) of the disc. In the case of functions vanishing at \(0\), the structure of the isometry is \(V(f\circ \sigma(z)-f\circ \sigma(0))\), for a surjective isometry \(V\) of \(E\). One of the techniques employed is to embed the analytic function spaces into the continuous function space, \({\mathcal C}_0(\Delta, E)\), and the other technique is to use, for real scalars, the results of \textit{G.-G. Ding} [Sci. China, Ser. A 45, No. 4, 479--483 (2002); Zbl 1107.46302] and \textit{L. Li} and \textit{R. Liu} [Tamsui Oxf. J. Math. Sci. 24, No. 4, 339--354 (2008; Zbl 1179.46008)] that allows certain surjective Lipschitz mappings of the unit ball to be extended to the whole space. The authors also show that, for a bounded Hermitian operator \(V\) of \(E\), \(f(z) \mapsto V(f(z))\) completely describes the structure of bounded Hermitian operators of \({\mathcal B}_0(\Delta,E)\).
spaces of vector-valued analytic finctions
strictly convex reflexive complex Banach spaces
scientific article; zbMATH DE number 6499613
scientific article