Abstract. We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.

Abstract. In this paper we state a Lipschitz version of a theorem due to Cambern concerning into linear isometries between spaces of vector-valued continuous functions and deduce a Lipschitz version of a celebrated theorem due to Jerison concerning onto linear isometries between such spaces.

a b s t r a c tWe characterize compact composition operators between different spaces of scalar-valued Lipschitz functions defined on metric spaces, not necessarily compact, and determine their spectra.

Abstract. Let (X, d) be a metric space, and let Lip(X) denote the Banach space of all scalar-valued bounded Lipschitz functions f on X endowed with one of the natural normswhere L( f ) is the Lipschitz constant of f . It is said that the isometry group of Lip(X) is canonical if every surjective linear isometry of Lip(X) is induced by a surjective isometry of X. In this paper we prove that if X is bounded separable and the isometry group of Lip(X) is canonical, then every 2-local isometry of Lip(X) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip(X) when X is bounded.

Given an infinite set Γ, we prove that the space of complex null sequences c 0 (Γ) satisfies the Mazur-Ulam property, that is, for each Banach space X, every surjective isometry from the unit sphere of c 0 (Γ) onto the unit sphere of X admits a (unique) extension to a surjective real linear isometry from c 0 (Γ) to X. We also prove that the same conclusion holds for the finite dimensional space ℓ m ∞ .2010 Mathematics Subject Classification. Primary 46B20, 46A22, 46B04, 46B25.