NEAR-ISOMETRIES BETWEEN $C(K)$-SPACES

Abstract

Let $X$, $Y$ be compact Hausdorff spaces and let $T:C(X,\mathbb{R})\to C(Y,\mathbb{R})$ be an invertible linear operator. Non-standard analysis is used to give a new intuitive proof of the Amir–Cambern result that if $\|T\|\hskip1pt\|T^{-1}\|\lt2$, then there is a homeomorphism $\psi:Y\to X$. The approach provides a proof of the following representation theorem for such near-isometries:

$$ Tf=(T1_X)(f\circ\psi)+Sf, $$

with $\|S\|\leq2(\|T\|-(1/\|T^{-1}\|))$, so $\|S\|\lt\|T\|$. If $\|T\|\hskip1pt\|T^{-1}\|=1$, then $S=0$, giving the well-known representation for isometries.