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Title: Banach-Stone Theorem for isometries on spaces of vector-valued differentiable functions
Authors: Leung, Denny H.
Ng, Hong Wai
Tang, Wee Kee
Keywords: Science::Mathematics
Issue Date: 2022
Source: Leung, D. H., Ng, H. W. & Tang, W. K. (2022). Banach-Stone Theorem for isometries on spaces of vector-valued differentiable functions. Journal of Mathematical Analysis and Applications, 514(1), 126305-.
Project: RG24/19(S)
Journal: Journal of Mathematical Analysis and Applications
Abstract: 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.
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2022.126305
Schools: School of Physical and Mathematical Sciences 
Rights: © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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