Extension of isometries on the unit sphere of L spaces.

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