On extension of isometries between unit spheres of \(L_{p}(\mu )\) and \(L_{p}(\nu ,H)\) (\(1< p \neq 2\), \(H\) is a Hilbert space)
An old and interesting question in the geometric theory of Banach spaces, due to \textit{D. Tingley} [Geom. Dedicata 22, 371--378 (1987; Zbl 0615.51005)] asks for conditions under which an onto isometry between the unit spheres extends to an affine or linear isometry between the spaces. Recently, a lot of work on problems of this kind was done by the Chinese school led by Ding Guanggui and his collaborators (see the reviews by this reviewer). In the present paper, the author introduces the notion of a \(p\)-affinely orthogonal map from an \(L_p\) space (\(p \neq 2\)) to a Banach space and shows that an \(p\)-affinely orthogonal map that is an isometry between the unit spheres extends to a linear isometry.
scientific article; zbMATH DE number 5077882