Remarks on Ritt operators, their $H^\infty$ functional calculus and associated square function estimates

Square functions for commuting families of Ritt operators

In this paper, we investigate the role of square functions defined for a $d$-tuple of commuting Ritt operators $(T_1,...,T_d)$ acting on a general Banach space $X$. Firstly, we prove that if the $d$-tuple admits a $H^\infty$ joint functional calculus, then it verifies various square function estimates. Then we study the converse when every $T_k$ is a $R$-Ritt operator. Under this last hypothesis, and when $X$ is a $K$-convex space, we show that square function estimates yield dilation of $(T_1,...,T_d)$ on some Bochner space $L_p(Ω;X)$ into a $d$-tuple of isomorphisms with a $C(\mathbb{T}^d)$ …

连接Ritt$_E$算子交换族的$H^\infty$-泛函演算与等距膨胀

Connecting $H^\infty$-functional calculus and isometric dilations for commuting families of Ritt$_E$ operators

本文证明，在UMD Banach空间上，Ritt$_E$算子交换族具有有界$H^\infty$-泛函演算当且仅当每个算子为$R$-Ritt$_E$且该族存在多项式有界的等距膨胀；在具有性质$(\alpha)$的空间中给出其他刻画。

Tadmor-Ritt算子的函数演算估计

Functional calculus estimates for Tadmor-Ritt operators

本文研究Tadmor-Ritt算子的$H^{\infty}$函数演算估计，改进了Vitse的结果，并探讨了离散平方函数估计对估计的影响。

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 by the…

与Ritt$_E$算子相关的平方函数

Square functions associated with Ritt$_E$ operators

本文针对单位圆子集定义的Ritt$_E$算子，引入二次泛函演算，证明在有限余型条件下其等价于有界泛函演算，并建立了相关平方函数的等价性。

The Ritt property of subordinated operators in the group case

Let $G$ be a locally compact abelian group, let $ν$ be a regular probability measure on $G$, let $X$ be a Banach space, let $π\colon G\to B(X)$ be a bounded strongly continuous representation. Consider the average (or subordinated) operator $S(π,ν) = \int_{G} π(t)\,dν(t)\,\colon X\to X$. We show that if $X$ is a UMD Banach lattice and $ν$ has bounded angular ratio, then $S(π,ν)$ is a Ritt operator with a bounded $H^\infty$ functional calculus. Next we show that if $ν$ is the square of a symmetric probability measure and $X$ is $K$-convex, then $S(π,ν)$ is a Ritt operator. We further show that …

On discrete subordination of power bounded and Ritt operators

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.

Dilation of Ritt operators on L^{p}-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 following s…

Square functions for Ritt operators on noncommutative $L^p$-spaces

For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_γ)$ where $B_γ$ is a Stolz domain. Moreover, we introduce the `column square functions' $\norm{x}_{T,c,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1}|T^{k-1}(I-T)^α(x)|^2\Big)^{1/2}}_{L^p(M)}$ and the `row square functions' $\norm{x}_{T,r,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1} |\Big(T^{k-1}(I-T)^α(x)\Big)^*|^2\Big)^{1/2}}_{L^p(M)}$ for any $α>0$ and any $x\in L^p(M)$. Then, we provide an example of Ritt operator which admits a completely bounde…

具有有限外围谱的算子的多边形函数演算

Polygonal functional calculus for operators with finite peripheral spectrum

本文研究 Banach 空间上外围谱有限且满足预解估计的有界算子，在正收缩或多项式有界加 R-有界条件下证明其具有有界多边形函数演算，并引入 Ritt_E 算子推广经典 Ritt 算子理论。

The $H^\infty$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators

In this paper we extend the $H^\infty$ functional calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called $S$-functional calculus. The $S$-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties. The $S$-functional calculus is based on the notion of $S$-spectru…

Remarks on functional calculus for perturbed first order Dirac operators

We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hytönen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in $L^p$ for $p$ in any open interval when suitable hypotheses are made. Hytönen and McIntosh then showed that $R$-bisectoriality in $L^p$ at one value of $p$ can be extrapolated in a neighborhood of $p$. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.

Resolvent representations for functions of sectorial operators

We obtain integral representations for the resolvent of $ψ(A)$, where $ψ$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $ψ$, we show that the operator $-ψ(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $ψ$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a pe…