Banach-Stone Theorem for isometries on spaces of vector-valued differentiable functions - Leung, Denny H. (2022)
Let Q⊆Rm, K⊆Rn be open sets, p,q∈N, 1≤r<∞ and let E,F be Banach spaces. Denote by C⁎p(Q,E)r the space of all f∈Cp(Q,E) with bounded derivatives of order ≤p, endowed with the norm ‖f‖=sups∈Q⁡‖[(‖∂λf(s)‖E)λ∈Λ]‖r, where ‖⋅‖r denotes the ℓr norm on RΛ, Λ={λ:|λ|≤p}. Let T:C⁎p(Q,E)r→C⁎q(K,F)r be a linear surjective isometry. Then m=n and p=q and there are a Cp-diffeomorphism τ:K→Q and Banach space isomorphisms V(t):E→F so that Tf(t)=V(t)f(τ(t)) if f∈C⁎p(Q,E),t∈K. The result holds in a more general setting. The proof establishes a direct link between isometries and biseparating maps.