Low rank compact operators and Tingley's problem

Let E and B be arbitrary weakly compact JB*-triples whose unit spheres are denoted by S(E) and S(B), respectively. We prove that every surjective isometry f : S(E) -> S(B) admits an extension to a surjective real linear isometry T : E -> B. This is a complete solution to Tingley's problem in the setting of weakly compact JB*-triples. Among the consequences, we show that if K(H, K) denotes the space of compact operators between arbitrary complex Hilbert spaces H and K, then every surjective isometry f : S(K(H, K)) -> S(K(H, K)) admits an extension to a surjective real linear isometry T : K(H, K) -> K(H, K). (C) 2018 Elsevier Inc. All rights reserved.