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arXiv 0801.2477math.FA
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加权复合算子的稳定性与不稳定性

Stability and instability of weighted composition operators

Jesus Araujo, Juan J. Font

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37 pages, 7 figures. A beamer presentation at http://www.araujo.tk
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摘要：

令$ε>0$。连续线性算子$T:C(X) \ra C(Y)$称为{ \em $ε$ -不交保持}，如果$\vc (Tf)(Tg)\vd_{\infty} \le ε$，只要$f,g\in C(X)$满足$\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$和$fg\equiv 0$ .在本文中，我们主要解决两个问题：1.加权复合算子与给定的$ε$ -不相交保持算子的距离有多近？2.-加权复合算子的集合离给定的$ε$ -不交保持算子有多远？我们解决这两个问题区分三种情况：$X$无限，$X$有限，$Y$单例（$ε$ -不相交保持泛函）。我们为这三种情况提供了精确的稳定性和不稳定性界限。

英文摘要：

Let $ε>0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $ε$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le ε$, whenever $f,g\in C(X)$ satisfy $\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$ and $fg\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $ε$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $ε$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($ε$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases.

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