The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

On isometric extension problem between two unit spheres

“…In [2], we generalized the result and got Let 0 < β n < 1 for all integers n. An operator V 0 is an isometric mapping from the unit sphere S(l βn ) onto S(l βn ), if and only if there exists a sequence {θ n } of signs and a permutation π of the integers, such that, for any integer n, β n = β π(n) and for any element x = ξ n e n ∈ S(l βn ), V 0 (x) = θ n ξ π(n) e n . Theorem 2.9.…”

“…We took another breakthrough. In [2], we began to consider the linear extension problem of 1-Lipschitz mappings between unit spheres. We got Theorem 5.1.…”

NONEXPANSIVE MAPPINGS AND EXPANSIVE MAPPINGS ON THE UNIT SPHERES OF SOMEF-SPACES

“…Ding [2] showed that every onto nonexpansive map between unit spheres of Hilbert spaces is an isometry and answered Tingley's problem affirmatively for Hilbert spaces. In recent work [11], the author proved that the only nonexpansive mappings from the unit sphere of L ∞ ( )-type spaces (including c 00 , c, ∞ ) onto the unit sphere of L ∞ ( ) are those arising from a bijection between and and a sign pattern.…”

“…However, what interests us is such maps defined only on the unit sphere, which can be connected with the isometric extension problem raised by Tingley in [12] and described as follows. [2] Nonexpansive mappings and expansive mappings on the unit spheres of some F-spaces 23…”

Extension of isometries on the unit sphere of L p spaces

“…In recent years, Ding and his students have been working on this topic and have obtained many important results (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Until now, the isometric extension problem for the surjective isometries between unit spheres of the same type classical Banach spaces has almost been solved.…”