Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum
Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. Let $1<p<\infty.$ In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all $x \in L_p(\mathcal{M})$, extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, the similar inequalities as above hold for arbitrary $L_\infty-L_1$ contractions with the same condition on the numerical range, yielding a vast generalisation of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. We further prove a variational inequality for contractively regular operators $T:L_p(\Omega)\to L_p(\Omega)$ whose peripheral spectrum is finite and satisfies a suitable resolvent estimate, extending earlier work of Le Merdy and Xu \cite{le-Merdy-Xu-q-variational-inequality}. Finally, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calder\'on and Below \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative $(p,p)$-maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalisation of Stein's method of embedding maximal function into analytic family of operators.