arXiv subjects

Jochen Glück

Publications and source records attributed to Jochen Glück.

At least 55 records · Page 3Linked to original sources

arXiv2020

On disjointness, bands and projections in partially ordered vector spaces

Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem - at first glance - intimately related to the lattice operations on $X$. The last fifteen years, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces. In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory. The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.

math.FA↗

arXiv2020

Uniform convergence of operator semigroups without time regularity

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $C_0$-semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $\mathbb{R}^d$, the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $C_0$-semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations - without any time regularity assumptions - by adapting the concept of the "semigroup at infinity", recently introduced by M.~Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e., powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

math.AP↗

arXiv2020

Positive irreducible semigroups and their long-time behaviour

The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as $t \to \infty$. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large $t$, while being absent for smaller times.

math.FA↗

arXiv2020

A note on lattice ordered $C^*$-algebras and Perron--Frobenius theory

A classical result of Sherman says that if the space of self-adjoint elements in a $C^*$-algebra $\mathcal{A}$ is a lattice with respect to its canonical order, then $\mathcal{A}$ is commutative. We give a new proof of this theorem which shows that it is intrinsically connected with the spectral theory of positive operator semigroups. Our methods also show that some important Perron--Frobenius like spectral results fail to hold in any non-commutative $C^*$-algebra.

math.OA↗

arXiv2020

Strict positivity for the principal eigenfunction of elliptic operators with various boundary conditions

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $Ω\subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge δ> 0$ for all $x \in \overlineΩ$, even if the boundary $\partial Ω$ is only Lipschitz continuous. Under such weak regularity assumptions the Hopf-Ole\uınik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map $L_p(Ω)$ into $C(\overlineΩ)$. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

math.AP↗

arXiv2020

Uniform convergence of stochastic semigroups

For stochastic $C_0$-semigroups on $L^1$-spaces there is wealth of results that show strong convergence to an equilibrium as $t \to \infty$, given that the semigroup contains a partial integral operator. This has plenty of applications to transport equations and in mathematical biology. However, up to now partial integral operators do not play a prominent role in theorems which yield uniform convergence of the semigroup rather than only strong convergence. In this article we prove that, for irreducible stochastic semigroups, uniform convergence to an equilibrium is actually equivalent to being partially integral and uniformly mean ergodic. In addition to this Tauberian theorem, we also show that our semigroup is uniformly convergent if and only if it is partially integral and the dual semigroup satisfies a certain irreducibility condition. Our proof is based on a uniform version of a lower bound theorem of Lasota and Yorke, which we combine with various techniques from Banach lattice theory.

math.FA↗

arXiv2020

On the decoupled Markov group conjecture

The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup $T = (T_t)_{t \in [0,\infty)}$ on $\ell^1$ has bounded generator if the operator $T_1$ is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite dimensional matrix semigroups of growing dimension - i.e., it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite dimensional systems. In this article we show that such attempts must necessarily fail, i.e., we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators $Q$ of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite dimensional Perron-Frobenius theory and Gelfand's $T = \operatorname{id}$ theorem.

math.FA↗

arXiv2020

Almost Interior Points in Ordered Banach Spaces and the Long--Term Behaviour of Strongly Positive Operator Semigroups

The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with the theory presented in the first part of the paper we deduce sufficiency criteria for such semigroups to converge (strongly or in operator norm) as time tends to infinity. This generalises known results for semigroups on Banach lattices as well as on normally ordered Banach spaces with unit.

math.FA↗

arXiv2020

A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

For a stationary moving average random field, a non-parametric low frequency estimator of the Lévy density of its infinitely divisible independently scattered integrator measure is given. The plug-in estimate is based on the solution $w$ of the linear integral equation $v(x) = \int_{\mathbb{R}^d} g(s) w(h(s)x)ds$, where $g,h:\mathbb{R}^d \rightarrow \mathbb{R}$ are given measurable functions and $v$ is a (weighted) $L^2$-function on $\mathbb{R}$. We investigate conditions for the existence and uniqueness of this solution and give $L^2$-error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.

math.ST↗

arXiv2019

Spectral gaps for hyperbounded operators

We consider a positive and power-bounded linear operator $T$ on $L^p$ over a finite measure space and prove that, if $TL^p \subseteq L^q$ for some $q > p$, then the essential spectral radius of $T$ is strictly smaller than $1$. As a special case, we obtain a recent result of Miclo who proved this assertion for self-adjoint ergodic Markov operators in the case $p=2$ and thereby solved a long-open problem of Simon and Høegh-Krohn. Our methods draw a connection between spectral theory and the geometry of Banach spaces: they rely on a result going back to Groh that encodes spectral gap properties via ultrapowers, and on the fact that an infinite dimensional $L^p$-space cannot by isomorphic to an $L^q$-space for $q \not= p$. We also prove a number of variations of our main result: (i) it follows from theorems of Lotz and Martínez that the condition $TL^p \subseteq L^q$ can be replaced with the weaker assumption that $T$ maps the positive part of the $L^p$-unit ball into a uniformly $p$-integrable set; (ii) while it is known that the positivity assumption on $T$ cannot in general be omitted, we show that we can replace it with the assumption that $T$ is contractive both on $L^p$ and on $L^q$; (iii) we prove a version of the theorem which allows us, under appropriate circumstances, to also consider non-finite measures spaces; (iv) our result also has a uniform version: there exists an upper bound $c \in [0,1)$ for the essential spectral radius of $T$, where $c$ depends on certain quantitative properties of $T$, $L^p$ and $L^q$.

math.SP↗

arXiv2019

Convergence of Positive Operator Semigroups

We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dominating a kernel operator converges strongly as $t \to \infty$. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.

math.FA↗

arXiv2018

Asymptotics of Operator Semigroups via the Semigroup at Infinity

We systematize and generalize recent results of Gerlach and Glück on the strong convergence and spectral theory of bounded (positive) operator semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an arbitrary commutative semigroup, and no topological assumptions neither on $S$ nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it. Next, we confine these abstract results to positive semigroups on Banach lattices with a quasi-interior point. In that situation, the said criteria are intimately linked to so-called AM-compact operators (which entail kernel operators and compact operators); and they imply that the original semigroup asymptotically embeds into a compact group of positive invertible operators on an atomic Banach lattice. By means of a structure theorem for such group representations (reminiscent of the Peter--Weyl theorem and its consequences for Banach space representations of compact groups) we are able to establish quite general conditions implying the strong convergence of the original semigroup. Finally, we show how some classical results of Greiner (1982), Davies (2005), Keicher (2006) and Arendt (2008) and more recent ones by Gerlach and Glück (2017) are covered and extended through our approach.

math.FA↗

arXiv2018

Eventually and asymptotically positive semigroups on Banach lattices

We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neumann operator on $L^2$ and the Laplacian with non-local boundary conditions on $L^2$ within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call "asymptotic positivity", where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as $t \to \infty$. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.

math.FA↗

arXiv2018

A short note on band projections in partially ordered vector spaces

Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I - P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.

math.FA↗

arXiv2018

A Toolkit for Constructing Dilations on Banach Spaces

We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if $X$ is a super-reflexive Banach space and $T$ is contained in the weakly closed convex hull of all invertible isometries on $X$, then $T$ admits a dilation to an invertible isometry on a Banach space $Y$ with the same regularity as $X$. The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.

math.FA↗

arXiv2018

Mean ergodicity vs weak almost periodicity

We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell^\infty$ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.

math.FA↗

arXiv2018

Long--Term Analysis of Positive Operator Semigroups via Asymptotic Domination

We consider positive operator semigroups on ordered Banach spac\-es and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups which are asymptotically dominated by $\mathcal{T}$. Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.

math.FA↗

arXiv2018

A criterion for the uniform eventual positivity of operator semigroups

Consider a $C_0$-semigroup $(e^{tA})_{t \ge 0}$ on a function space or, more generally, on a Banach lattice $E$. We prove a sufficient criterion for the operators $e^{tA}$ to be positive for all sufficiently large times $t$, while the semigroup itself might not be positive. This complements recently established criteria for the individual orbits of the semigroup to become eventually positive for all positive initial values. We apply our main result to study the qualitative behaviour of the solutions to various partial differential equations.

math.FA↗