论文标题
具有有限外围光谱的操作员的多边形功能演算
Polygonal functional calculus for operators with finite peripheral spectrum
论文作者
论文摘要
令$ t \ colon x \ to x $为Banach空间上的有界运算符，其频谱$σ（t）$包含在封闭的单元光盘$ \ overline {\ mathbb d} $中。假设外围光谱$σ（t）\ cap {\ mathbb t} $是有限的，并且$ t $满足分辨率估算$$ \ vert（z-t）^{ - 1} \ vert \ vert \ simsim \ max \ max \ max \ max \ max \ big big big big { σ（t）\ cap {\ mathbb t} \ bigr \}，\ qquad z \ in \ intlline {\ mathbb d}^c。$$我们证明，$ t $允许一个有界的多角函数计算，即估计$ \ vert（t）\ vert（t）对于某些多边形$Δ\ subset {\ Mathbb d} $，在以下两种情况下，在每种情况下，（i）$ x = l^p $ col，sup \ {\ vertϕ（z）\ vert \，：\，z \inδ\} $，在以下两种情况下，所有$ x = l^p $ col col of $ x = l^p <p <p <p <p <p <p <p <p <p <p <p <p < l^p $是一个积极的收缩； （ii）或$ t $在σ（t）\ cap {\ mathbb t}中的所有$ξ\中是多项式界限\ edline {\ mathbb d}^c \} $是$ r $ bunded（$ x $是任意的）。这两个结果中的每一个都扩展了关于希尔伯特空间中多边形功能积分的de laubenfels定理。我们的调查需要介绍，对于任何有限的集合$ e \ subset {\ mathbb t} $，是RITT $ _e $ operator的概念，该概述将Ritt Operator的经典概念概述。我们研究这些Ritt $ _e $运算符及其自然功能演算。
Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum $σ(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assume that the peripheral spectrum $σ(T)\cap{\mathbb T}$ is finite and that $T$ satisfies a resolvent estimate $$\Vert(z-T)^{-1}\Vert\lesssim \max\bigl\{\vert z -ξ\vert^{-1}\, :\,ξ\in σ(T)\cap{\mathbb T}\bigr\}, \qquad z\in\overline{\mathbb D}^c.$$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $\Vertϕ(T)\Vert\lesssim \sup\{\vertϕ(z)\vert\, :\, z\inΔ\}$ for some polygon $Δ\subset{\mathbb D}$ and all polynomials $ϕ$, in each of the following two cases : (i) either $X=L^p$ for some $1<p<\infty$, and $T\colon L^p\to L^p$ is a positive contraction; (ii) or $T$ is polynomially bounded and for all $ξ\in σ(T)\cap{\mathbb T},$ there exists a neighborhood $\mathcal V$ of $ξ$ such that the set $\{(ξ-z)(z-T)^{-1}\, :\, z\in{\mathcal V}\cap \overline{\mathbb D}^c\}$ is $R$-bounded (here $X$ is arbitrary). Each of these two results extends a theorem of de Laubenfels concerning polygonal functional calculus on Hilbert space. Our investigations require the introduction, for any finite set $E\subset{\mathbb T}$, of a notion of Ritt$_E$ operator which generalises the classical notion of Ritt operator. We study these Ritt$_E$ operators and their natural functional calculus.
