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

Proceedings of the Edinburgh Mathematical Society

N. J. Cutland¹ 

[1] G. B. Zimmer

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

DOI  :  10.1017/S001309150300049X

学科分类：数学（综合）

来源: Cambridge University Press

【 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.

【 授权许可】

Unknown