Michael Lin

We study the asymptotic behavior of Markov operators $P_μ$ defined by convolution with a probability measure $μ$ on the unit circle $\mathbb T$. We prove that when $μ$ is adapted, $P_μ$ satisfies Doeblin's condition if and only if some power $μ^k$ is non-singular. We give an example of a symmetric probability measure $μ$ on $\mathbb T$, such that the reversible stationary chain induced by $P_μ$ is $ρ$-mixing, but $P_μ$ does not satisfy Doeblin's condition. We look at the spectra of $P_μ$ in the different $L_p$ spaces when $P_μ$ is, or is not, $ρ$-mixing.

Let $ T $ be the Koopman operator of a measure-preserving transformation $ \theta $ of a probability space $ (X,\Sigma,\mu) $. We study the convergence properties of the averages $ M_nf: = \frac1n\sum_{k = 0}^{n-1}T^kf $ when $ f \in L^r(\mu) $, $ 0<r<1 $. We prove that if $ \int |M_nf|^r d\mu \to 0 $, then $ f \in \overline{(I-T)L^r} $, and show that the converse fails whenever $ \theta $ is ergodic aperiodic. When $ \theta $ is invertible ergodic aperiodic, we show that for $ 0<r<1 $ there exists $ f_r \in (I-T)L^r $ for which $ M_nf_r $ does not converge a.e. (although $ \int |M_nf|^r d\mu \to 0 $). We further establish that for $ 1 \leq p <\frac{1}{r}, $ there is a dense $ G_\delta $ subset $ { \mathscr F}\subset L^p(X,\mu) $ such that $ \limsup_n \frac{|T^nh|}{n^r} = \infty $ a.e. for any $ h \in { \mathscr F} $. An appendix by Guy Cohen deals with rates of convergence of $ \frac{T^ng}n \to 0 $ for $ g\in L^1(X,\mu) $. When $ T $ is induced by an irrational rotation of $ \mathbb T $, the Hardy spaces $ H^r(\mathbb T) $ are $ T $-invariant. For $ 0<r<1 $, we prove that $ H^r(\mathbb T) = \{\text{constants}\}\oplus \overline{(I-T)H^r(\mathbb T)} $, and $ \int |M_nf|^r d\mu \to 0 $ for $ f \in \overline{(I-T)H^r(\mathbb T)} $. However, there exists $ f\!\in\!(I-T)H^r(\mathbb T) $ such that $ M_nf $ does not converge a.e.

Let $T$ be the Koopman operator of a measure preserving transformation $θ$ of a probability space $(X,Σ,μ)$. We study the convergence properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in L^r(μ)$, $0<r<1$. We prove that if $\int |M_nf|^r dμ\to 0$, then $f \in \overline{(I-T)L^r}$, and show that the converse fails whenever $θ$ is ergodic aperiodic. When $θ$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\int |M_nf|^r dμ\to 0$). We further establish that for $1 \leq p <\frac{1}{r},$ there is a dense $G_δ$ subset ${\mathcal F}\subset L^p(X,μ)$ such that $\limsup_n \frac{|T^nh|}{n^r}=\infty$ a.e. for any $h \in {\mathcal F}$.

Let $T$ be a bounded linear operator on a Banach space $X$ satisfying $\|T^n\|/n \to 0$. We prove that $T$ is uniformly ergodic if and only if the one-sided ergodic Hilbert transform $H_Tx:= \lim_{n\to\infty} \sum_{k=1}^n k^{-1}T^k x$ converges for every $x \in \overline{(I-T)X}$. When $T$ is power-bounded (or more generally $(C,α)$ bounded for some $0< α<1$), then $T$ is uniformly ergodic if and only if the domain of $H_T$ equals $(I-T)X$. We then study rotational uniform ergodicity -- uniform ergodicity of every $λT$ with $|λ|=1$, and connect it to convergence of the rotated one-sided ergodic Hilbert transform, $H_{λT}x$. In the Appendix we prove that positive isometries with finite-dimensional fixed space on infinite-dimensional Banach lattices are never uniformly ergodic. In particular, the Koopman operators of ergodic, even non-invertible, probability preserving transformations on standard spaces are never uniformly ergodic.

Following Berm\'udez et al. (ArXiv: 1706.03638v1), we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Ces\`aro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Ces\`aro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\|T^n\|=O(n/\sqrt {\log n})$; if $T$ is absolutely Ces\`aro bounded, $\|T^n\|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\|T^n\|=O((\log n)^\kappa)$ for some $\kappa >0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Ces\`aro means of order $\alpha$ converge strongly when $\alpha >1$.