Extension of isometries on the unit sphere of L p spaces

Abstract

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp(µ) (1 < p < ∞, p ≠ 2) and a Banach space E can be extended to a linear isometry from Lp(µ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(µ), then E is linearly isometric to Lp(µ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp(µ1, H1) and Lp(µ2, H2) must be an isometry and can be extended to a linear isometry from Lp(µ1, H1) to Lp(µ2, H2), where H1 and H2 are Hilbert spaces.