arXiv subjects

Tanja Eisner

Publications and source records attributed to Tanja Eisner.

At least 19 recordsLinked to original sources

arXiv2026

Weak limit semigroup in operator theory and ergodic theory

We study the weak limit semigroup of an operator $T$, i.e., the set of all operators being weak limit points of the powers of $T$, in three different but related contexts: Koopman operators of measure-preserving transformations, contractions/isometries/unitaries on separable Hilbert spaces and positive operators on $L^p$-spaces. Hereby we focus on finding large subsets of the weak limit semigroup, in particular in the generic case.

math.FA↗

arXiv2025

A generic transformation is invertible

We show that, on a standard non-atomic probability space, invertible measure-preserving transformations form a dense $G_\delta$ subset of the space of all measure-preserving transformations endowed with the strong (=weak) operator topology. This implies that all properties which are generic for invertible transformations are also generic for general ones. We further show that invertible Koopman operators form a dense $G_\delta$ subset of all bi-stochastic operators for the weak operator topology, and the same holds for general Koopman operators.

math.DS↗

arXiv2022

A view on multiple recurrence

In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemer\'edi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition and properties of the Gowers-Host-Kra uniformity seminorms.

math.DS↗

arXiv2021

Divergence of weighted square averages in $L^1$

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial $P\in \mathbb{Z}[\cdot]$, consider the set of all $θ\in[0,1)$ such that for every aperiodic system $(X,μ, T)$ there is a function $f\in L^1(X,μ)$ such that the weighted averages along squares $$ {\frac{1}{N}\sum_{n=1}^N} e(P(n)θ)T^{n^2}f $$ diverge on a set with positive measure. We show that this set is residual and includes the rational numbers as well as a dense set of Liouville numbers. This on one hand extends the divergence result for squares in $L^1$ of the first author and Mauldin and on the other hand shows that the convergence result for linear weights for squares due to the second author and Krause in $L^p$, $p>1$ does not hold for $p=1$.

math.DS↗

arXiv2021

Embeddability of real and positive operators

Embedding discrete Markov chains into continuous ones is a famous open problem in probability theory with many applications. Inspired by recent progress, we study the closely related questions of embeddability of real and positive operators into real or positive $C_0$-semigroups, respectively, on finite and infinite-dimensional separable sequence spaces. For the real case we give both sufficient and necessary conditions for embeddability. For positive operators we present necessary conditions for positive embeddability including a full description for the $2\times 2$-case. Moreover, we show that real embeddability is topologically typical for real contractions on $\ell^2$.

math.FA↗

arXiv2020

Resolvent conditions and growth of powers of operators

Following Bermúdez 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 Cesàro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Cesàro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\|T^n\|=O(n/\sqrt {\log n})$; if $T$ is absolutely Cesàro bounded, $\|T^n\|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\|T^n\|=O((\log n)^κ)$ for some $κ>0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order $α$ converge strongly when $α>1$.

math.DS↗

arXiv2020

Power bounded operators and the mean ergodic theorem for subsequences

Let $T$ be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages $N^{-1}\sum_{n=1}^N T^{a_n}$ converge in the strong operator topology for a wide range of sequences $(a_n)$, including the integer part of most of subpolynomial Hardy functions. Moreover, we show that the weighted averages $N^{-1}\sum_{n=1}^N e^{2πi g(n)}T^{a_n}$ also converge for many reasonable functions $g$. In particular, we generalize the polynomial mean ergodic theorem for power bounded operators due to ter Elst and the second author \cite{tEM} to real polynomials and polynomial weights.

math.FA↗

arXiv2019

Nilsystems and ergodic averages along primes

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao we observe everywhere convergence of such averages for nilsystems and continuous functions.

math.DS↗

arXiv2018

A polynomial version of Sarnak's conjecture

Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in $L^q$, q>1, we introduce polynomial versions of the Sarnak conjecture for minimal systems.

math.DS↗

arXiv2018

Automatic sequences as good weights for ergodic theorems

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.

math.DS↗

arXiv2017

On modulated ergodic theorems

Let $T$ be a weakly almost periodic (WAP) linear operator on a Banach space $X$. A sequence of scalars $(a_n)_{n\ge 1}$ {\it modulates} $T$ on $Y \subset X$ if $\frac1n\sum_{k=1}^n a_kT^k x$ converges in norm for every $x \in Y$. We obtain a sufficient condition for $(a_n)$ to modulate every WAP operator on the space of its flight vectors, a necessary and sufficient condition for (weakly) modulating every WAP operator $T$ on the space of its (weakly) stable vectors, and sufficient conditions for modulating every contraction on a Hilbert space on the space of its weakly stable vectors. We study as an example modulation by the modified von Mangoldt function $Λ'(n):=\log n1_\mathbb P(n)$ (where $\mathbb P =(p_k)_{k\ge 1}$ is the sequence of primes), and show that, as in the scalar case, convergence of the corresponding modulated averages is equivalent to convergence of the averages along the primes $\frac1n\sum_{k=1}^n T^{p_k}x$. We then prove that for any contraction $T$ on a Hilbert space $H$ and $x \in H$, and also for every invertible $T$ with $\sup_{n \in \mathbb Z} \|T^n\| <\infty$ on $L^r(Ω,μ)$ ($1<r< \infty$) and $f \in L^r$, the averages along the primes converge.

math.FA↗

arXiv2017

On the pointwise entangled ergodic theorem

We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$ \sum_{n=1}^N T_a^n A_{a-1}T_{a-1}^nA_{a-1}\cdot \ldots \cdot A_0 T_0^nf$$ for $f\in L^p(X,μ)$, $p\geq 1$. We also discuss examples and present a continuous version of the result.

math.DS↗

arXiv2016

Wiener's lemma along primes and other subsequences

Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $\mu$ on the unit circle via the behavior of its Fourier coefficients $\hat\mu(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results of Goldstein and Goldstein, Nagy. The results are complemented by some open questions and indication of interesting research directions. After this paper had been published, it was pointed out to us by Emmanuel Lesigne and M\'at\'e Wierdl that there is a gap in the Example on return times sequences along polynomials on page 13. Indeed, to make the argument there work, one needs a Wiener-Wintner type result for polynomial averages with a precise information about the limit, and this is presently out of reach.

math.FA↗

arXiv2014

(Uniform) Convergence of Twisted Ergodic Averages

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,Σ,μ)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: \[ \{\frac{1}{N} \sum_{n\leq N} e(p(n)) T^{n}f(x) \} \] and for "twisted" polynomial ergodic averages: \[ \{\frac{1}{N} \sum_{n\leq N} e(n θ) T^{P(n)}f(x) \} \] for certain classes of badly approximable $θ\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $μ$-a.e. for $f \in L^p(X), \ p >1,$ and arbitrary $θ\in [0,1]$.

math.DS↗

arXiv2014

Rigidity of contractions on Hilbert spaces

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times the identity operator in the strong limit set of its powers, while $T^{n_j}$ converges weakly to zero along a sequence $\{n_j\}$ with density one. The continuous analogue is presented for isometric ang unitary $C_0$-(semi)groups.

math.FA↗

arXiv2013

Linear sequences and weighted ergodic theorems

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form (g(S^ny)) for a measure preserving system (Y,S) and g\in L^\infty(Y), avoiding in the latter case the problem of finding the full measure set of appropriate points y.

math.DS↗

arXiv2012

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.

math.DS↗

arXiv2012

On the Wiener-Wintner theorem for nilsequences

We give a Fourier analytic proof of the generalisation due to Host and Kra of the classical Wiener-Wintner theorem and give some explicit bounds on the limit of the weighted ergodic averages.

math.DS↗