Extension of isometries on the unit sphere of spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of (A mu) (1 < < a, not equal 2) and a Banach space can be extended to a linear isometry from (A mu) onto , which means that if the unit sphere of is (metrically) isometric to the unit sphere of (A mu), then is linearly isometric to (A mu). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of (A mu(1), (1)) and (A mu(2), (2)) must be an isometry and can be extended to a linear isometry from (A mu(1), (1)) to (A mu(2), (2)), where (1) and (2) are Hilbert spaces.