Banach space operators with a bounded H∞ functional calculus
Banach space operators with a bounded H∞ functional calculus
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DOI：
10.1017/s1446788700037393
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发表时间：
1996-02
期刊：
影响因子：
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通讯作者：
M. Cowling;I. Doust;Alan Micintosh;A. Yagi
中科院分区：
文献类型：
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作者：
M. Cowling;I. Doust;Alan Micintosh;A. Yagi
本文给出了f（T）的一般定义，当T是作用在Banach空间中的线性算子，其谱位于某个扇区内，且满足一定的预解式界，以及当f在较大扇区上全纯时。我们还研究了如何某些属性的功能演算，如存在一个有界的H∈功能演算，虚功率的界限，平方函数估计。特别地，我们证明了，如果T是在自反Lp空间，那么T有界H∈功能演算当且仅当T和它的对偶满足平方函数估计。给出的例子表明，在Hilbert空间中的运营商持有的一些定理不扩展到一般的Banach空间设置。
Abstract In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector. We also examine how certain properties of this functional calculus, such as the existence of a bounded H∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive Lp space, then T has a bounded H∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.