Hyers-Ulam stability of ε-isometries between the positive cones of Lp-spaces

Abstract

Let (Ωi,Σi,μi), i=1,2, be two measure spaces, 1<p,q<∞ with 1p+1q=1, Xi=Lp(Ωi,Σi,μi), Yi=Lq(Ωi,Σi,μi), and let Xi+={f∈Xi:f≥0a.e.} be the positive cone of Xi. In this paper, we first show a weak stability formula of a standard ε-isometry F:X1+→X2+: For every x⁎∈Y1+, there exists a unique ϕ∈Y2+ with ‖x⁎‖=‖ϕ‖≡r such that|〈x⁎,u〉−〈ϕ,F(u)〉|≤2rε,for allu∈X1+. Making use of it, we show the following Hyers-Ulam stability of ε-isometry F:X1+→X2+: If F is almost surjective, then there exists a unique additive surjective isometry V:X1+→X2+ (the restriction of a linear surjective isometry between X1 and X2) defined as V(u)=limt→∞⁡F(tu)t for each u∈X1+ so that‖F(u)−V(u)‖≤4ε,for allu∈X1+.