2-local isometries on spaces of differentiable functions (Researches on isometries as preserver problems and related topics)

Abstract

Let C^{(2)}([0, 1]) be the Banach space of 2-times continuously differentiable functions on the closed unit interval [0, 1] equipped with the norm Vert fVert_{sigma}= |f(0)|+|f'(0)|+Vert f"Vert_{infty}, where Vert gVert_{infty}=sup{|g(t)| : tin[0, 1]} for g. If T : (C^{(2)}([0, 1]), Vert Vert_{sigma})arrow(C^{(2)}([0, 1]), Vert Vert_{sigma}) is a 2-local isometry, then T is a surjective complex-linear isometry.