By means of a new technique, we develop further a discrete subordination approach to the functional calculus of power bounded and Ritt operators initiated by N. Dungey in [19]. This allows us to show, in particular, that (infinite) convex combinations of powers of Ritt operators are Ritt. Moreover, we provide a unified framework for several main results on discrete subordination from [19] and answer a question left open in [19]. The paper can be considered as a complement to [26] for the discrete setting.
Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators
Aix-Marseille University（艾克斯-马赛大学）；CNRS, I2M Marseille（法国国家科学研究中心，I2M马赛）；Nicolaus Copernicus University（尼古拉·哥白尼大学）；Institute of Mathematics Polish Academy of Sciences（波兰科学院数学研究所）
$H^\infty$ Functional Calculus for a Commuting Pair of $\text{Ritt}_{\text{E}}$ Operators
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
设T：X --> X $ be a power bounded operator on Banach space. An operator C : X --> Y$对T是容许的，如果它满足一个估计$\sum_k\norm{CT^k(x)}^2\,\leq M^2\norm{x}^2$ .继Harper和Wynn之后，我们研究了一个Weiss猜想在这个离散环境中的有效性。证明了当X是自反的且T是满足适当平方函数估计的Ritt算子时，C对T是可容许的当且仅当它对$ω\in \Cdb$，$|ω|<1$满足一致估计$(1-| ω|^2)^{1/2}\norm{C(I-ωT)^{-1}}\,\leq K$.我们将这一结果推广到更一般的α-容许设置。然后，我们调查一个自然的变体涉及R-有界的容许性，并提供了我们的一般结果适用的例子。
H^\infty functional calculus and square function estimates for Ritt operators
A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p< \infty , we study the validity of square functions estimates Norm{( \sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^ \infty -calculus on $Lp$ -spaces …
For any Ritt operator T:L^{p}(Ω) --> L^{p}(Ω), for any positive real number α, and for any x in L^{p}, we consider the square functions |x |_{T,α} = \Bigl | \Bigl ( \sum_{k=1}^{\infty} k^{2α-1} \bigl |T^{k-1}(I-T)^αx \bigr |^2 \Bigr )^{1/2}_{L^{p}}. We show that if T is actually an R-Ritt operator, then these square functions are pairwise equivalent. Then we show that T and its adjoint T* acting on L^{p'} both satisfy uniform estimates |x| _{T,1} \lesssim |x| _{L^{p}} and |y| _{T*,1} \lesssim |y| _{L^{p'}} for x in L^{p} and y in L^{p'} if and only if T is R-Ritt and admits a dilation in the f…
