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

[摘要] Let $X$, $Y$ be compact Hausdorff spaces and let $T:C(X,mathbb{R})o C(Y,mathbb{R})$ 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 $psi:Yo X$. The approach provides a proof of the following representation theorem for such near-isometries:$$ Tf=(T1_X)(fcircpsi)+Sf, $$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.

[学科分类] 数学（综合）

[关键词] Primary 47B38;Secondary 46S20;Banach–Stone;non-standard analysis;representation theorem