Axioms (Feb 2024)

On Coarse Isometries and Linear Isometries between Banach Spaces

  • Yuqi Sun

DOI
https://doi.org/10.3390/axioms13030157
Journal volume & issue
Vol. 13, no. 3
p. 157

Abstract

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Let X,Y be two Banach spaces and f:X→Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a Lp(1p∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.

Keywords