Isometries of $H^{p}$ spaces of the torus

Abstract:

Denote by ${H^p}(1 \leqq p \leqq \infty )$ the Banach spaces of complex-valued functions in ${L^p}$ of the torus whose Fourier coefficients vanish off a half plane determined by a lexicographic ordering. The surjective isometries of the spaces ${H^p}(p \ne 2)$ are characterized in terms of unimodular functions on the circle and conformal maps of the disc. For $1 < p < \infty (p \ne 2)$ the proof depends upon a characterization of certain invariant subspaces previously given by the authors.