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Markus Haase

Publications and source records attributed to Markus Haase.

At least 19 recordsLinked to original sources
arXiv2025

Precompactness notions in Kaplansky--Hilbert modules and extensions with discrete spectrum

This paper is a continuation of our work on the functional-analytic core of the classical Furstenberg-Zimmer theory. We introduce and study (in the framework of lattice-ordered spaces) the notions of total order-boundedness and uniform total order-boundedness. Either one generalizes the concept of ordinary precompactness known from metric space theory. These new notions are then used to define and characterize "compact extensions" of general measure-preserving systems (with no restrictions on the underlying probability spaces nor on the acting groups). In particular, it is (re)proved that compact extensions and extensions with discrete spectrum are one and the same thing. Finally, we show that under natural hypotheses a subset of a Kaplansky-Banach module is totally order bounded if and only if it is cyclically compact (in the sense of Kusraev).

math.DS↗
arXiv2024

Vector-Valued Holomorphic Functions and Abstract Fubini-Type Theorems

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in \Omega$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(\Omega)$, $1\le p \le \infty$. It is proved (among other things) that \[ \langle t\mapsto \varphi( f(\cdot,t) ) , \mu \rangle= \varphi(z \mapsto \langle f(z, \cdot) , \mu\rangle )\] whenever $\mu \in E'$ and $\varphi$ is a linear functional on $H^\infty(O)$ that is sequentially continuous with respect to bounded pointwise convergence in $H^\infty(O)$.

math.FA↗
arXiv2024

Reactive additive capillary stamping with double network hydrogel-derived aerogel stamps under solvothermal conditions

Integration of solvothermal reaction products into complex thin-layer architectures is frequently achieved by combinations of layer transfer and subtractive lithography, whereas direct additive substrate patterning with solvothermal reaction products has remained challenging. We report reactive additive capillary stamping under solvothermal conditions as a parallel contact-lithographic access to patterns of solvothermal reaction products in thin-layer configurations. To this end, corresponding precursor inks are infiltrated into mechanically robust mesoporous aerogel stamps derived from double-network hydrogels (DNHGs). The stamp is then brought into contact with a substrate to be patterned under solvothermal reaction conditions inside an autoclave. The precursor ink forms liquid bridges between the topographic surface pattern of the stamp and the substrate. Evaporation-driven enrichment of the precursors in these liquid bridges along with their liquid-bridge-guided conversion into the solvothermal reaction products yields large-area submicron patterns of the solvothermal reaction products replicating the stamp topography. As example, we prepared thin hybrid films, which contained ordered monolayers of superparamagnetic submicron nickel ferrite dots prepared by solvothermal capillary stamping surrounded by nickel electrodeposited in a second, orthogonal substrate functionalization step. The submicron nickel ferrite dots acted as magnetic hardener halving the remanence of the ferromagnetic nickel layer. In this way, thin-layer electromechanical systems, transformers and positioning systems may be customized.

physics.app-ph↗
arXiv2024

Solvent-Free High-Temperature Capillary Stamping of Stimuli-Responsive Polymers: Wettability Management by Orthogonal Substrate Functionalization

The wettability of surfaces determines their antifouling, antifogging, anti-icing, and self-cleaning properties as well as their usability for sensing, oil-water separation, water collection, and water purification. Solvent-free high-temperature capillary stamping of stimuli-responsive polymers yielding arrays of stimuli-responsive polymer microdots on differently modified substrates enables the flexible generation of switchable surfaces with different water contact angles (WCAs). Potential problems associated with the deposition of polymer solutions, such as the handling of volatile organic solvents, phase separation induced by solvent evaporation, and capillarity-driven flow processes, are circumvented. We used composite stamps with topographically patterned contact surfaces consisting of metallic nickel cores and porous MnO2 coatings taking up the stimuli-responsive polymers. The short transport paths from the MnO2 contact layers to the counterpart substrates enabled the stamping of polymer melts containing components impeding flow, such as carbon nanotubes (CNTs). Thus-obtained arrays of polymer-CNT hybrid microdots prevent problems associated with continuous coatings including delamination and crack propagation. Moreover, the range within which the properties of the stamped stimuli-responsive polymer microdots are switchable can be tuned by orthogonal substrate modification. As an example, we stamped hybrid microdots consisting of poly(2-(methacryloyloxy)ethyl ferrocenecarboxylate) (PFcMA) and CNTs onto indium tin oxide (ITO) substrates. Coating the ITO substrates with a poly(ethylene oxide)-terminated silane shifted the WCAs obtained by switching the PFcMA between its oxidized and reduced states by nearly 50{\deg}.

cond-mat.soft↗
arXiv2022

Holomorphic H\"ormander-Type Functional Calculus on Sectors and Strips

In this paper, recent abstract multiplier theorems for $0$-sectorial and $0$-strip type operators by Kriegler and Weis (2018) are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic H\"ormander-type functions on sectors and strips are introduced with a scale of smoothness being finer than the classical polynomial one. Moreover, we establish alternative descriptions of these spaces involving Schwartz and "holomorphic Schwartz" functions. Finally, the abstract results are combined with a recent result by Carbonaro and Dragi\v{c}evi\'c (2017) to obtain an improvement -- with respect to the smoothness condition -- of the known H\"ormander-type multiplier theorem for general symmetric contraction semigroups.

math.FA↗
arXiv2021

Phenolic Resin Dual-Use Stamps for Capillary Stamping and Decal Transfer Printing

We report an optimized two-step thermopolymerization process carried out in contact with micropatterned molds that yields porous phenolic resin dual-use stamps with topographically micropatterned contact surfaces. With these stamps, two different parallel additive substrate manufacturing methods can be executed: capillary stamping and decal transfer microlithography. Under moderate contact pressures, the porous phenolic resin stamps are used for non-destructive ink transfer to substrates by capillary stamping. Continuous ink supply through the pore systems to the contact surfaces of the porous phenolic resin stamps enables multiple successive stamp-substrate contacts for lithographic ink deposition under ambient conditions. No deterioration of the quality of the deposited pattern occurs and no interruptions for ink replenishment are required. Under high contact pressure, porous phenolic resin stamps are used for decal transfer printing. In this way, the tips of the stamps' contact elements are lithographically transferred to counterpart substrates. The granular nature of the phenolic resin facilitates the rupture of the contact elements upon stamp retraction. The deposited phenolic resin micropatterns characterized by abundance of exposed hydroxyl groups are used as generic anchoring sites for further application-specific functionalizations. As example, we deposited phenolic resin micropatterns on quartz crystal microbalance resonators and further functionalized them with polyethylenimine for preconcentration sensing of humidity and gaseous formic acid. We envision that also preconcentration coatings for other sensing methods, such as attenuated total reflection infrared spectroscopy and surface plasmon resonance spectroscopy, are accessible by this functionalization algorithm

cond-mat.mtrl-sci↗
arXiv2021

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.

math.DS↗
arXiv2021

Slippery Polymer Monoliths: Surface Functionalization with Ordered MoS2 Microparticle Arrays

Components of technical systems and devices often require self-lubricating properties, which are implemented by means of dry lubricants. However, continuous lubricant coatings on the components' surfaces often suffer from poor adhesion, delamination and crack propagation. The replacement of continuous coatings with dense ordered arrays of microparticles consisting of dry lubricants may overcome these drawbacks. Using the well-established solid lubricant MoS2 as model system, we demonstrate that the sliding capability of polymeric monoliths can be significantly enhanced by integration of arrays of micron-sized dry lubricant microparticles into their contact surfaces. To synthesize the MoS2 microparticle arrays, we first prepared ordered hexagonal arrays of ammonium tetrathiomolybdate (ATM) microparticles on Si wafers by molding against poly(dimethylsiloxane) templates followed by high-temperature conversion of the ATM microparticles into MoS2 microparticles under Ar/H2 atmosphere in the presence of elemental sulfur. Finally, the obtained large-scale hexagonal MoS2 microparticle arrays were transferred to the surfaces of polymer monoliths under conservation of the array ordering. Self-lubrication of components of technical systems by incorporation of dry lubricant microparticle arrays into their contact surfaces is an example for overcoming the drawbacks of continuous functional coatings by replacing them with microparticle arrays.

physics.app-ph↗
arXiv2020

On the Fundamental Principles of Unbounded Functional Calculi

In this paper, a new axiomatization for unbounded functional calculi is proposed and the associated theory is elaborated comprising, among others, uniqueness and compatibility results and extension theorems of algebraic and topological nature. In contrast to earlier approaches, no commutativity assumptions need to be made about the underlying algebras. In a second part, the abstract theory is illustrated in familiar situations (sectorial operators, semigroup generators, normal operators). New topological extension theorems are proved for the sectorial calculus and the Hille--Phillips calculus. Moreover, it is shown that the Stieltjes and the Hirsch calculus for sectorial operators are subcalculi of a (small) topological extension of the sectorial calculus.

math.FA↗
arXiv2020

The Functional Calculus Approach to the Spectral Theorem

A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators, here the functional calculus is not treated as a subordinate but as the central concept. Based on five simple axioms for a "measurable functional calculus", the theory of such calculi is developed in detail, including spectral theory, uniqueness results and construction principles. Finally, the functional calculus form of the spectral theorem is stated and proved, with some proof variants being discussed.

math.FA↗
arXiv2018

Asymptotics of Operator Semigroups via the Semigroup at Infinity

We systematize and generalize recent results of Gerlach and Gl\"uck 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 Gl\"uck (2017) are covered and extended through our approach.

math.FA↗
arXiv2017

On systems with quasi-discrete spectrum

In this paper we re-examine the theory of systems with quasi-discrete spectrum initiated in the 1960's by Abramov, Hahn, and Parry. In the first part, we give a simpler proof of the Hahn--Parry theorem stating that each minimal topological system with quasi-discrete spectrum is isomorphic to a certain affine automorphism system on some compact Abelian group. Next, we show that a suitable application of Gelfand's theorem renders Abramov's theorem --- the analogue of the Hahn-Parry theorem for measure-preserving systems --- a straightforward corollary of the Hahn-Parry result. In the second part, independent of the first, we present a shortened proof of the fact that each factor of a totally ergodic system with quasi-discrete spectrum (a "QDS-system") has again quasi-discrete spectrum and that such systems have zero entropy. Moreover, we obtain a complete algebraic classification of the factors of a QDS-system. In the third part, we apply the results of the second to the (still open) question whether a Markov quasi-factor of a QDS-system is already a factor of it. We show that this is true when the system satisfies some algebraic constraint on the group of quasi-eigenvalues, which is satisfied, e.g., in the case of the skew shift.

math.DS↗
arXiv2016

Functional calculus on real interpolation spaces for generators of $C_{0}$-groups

We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded $H^{\infty}_{1}$-calculus on real interpolation spaces. Additional results are derived from this.

math.FA↗
arXiv2015

Form Inequalities for Symmetric Contraction Semigroups

Consider --- for the generator \({-}A\) of a symmetric contraction semigroup over some measure space $\mathrm{X}$, $1\le p < \infty$, $q$ the dual exponent and given measurable functions $F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ --- the statement: $$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ {\em for all $\mathbb{C}^d$-valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$.} It is shown that this statement is valid in general if it is valid for $\mathrm{X}$ being a two-point Bernoulli $(\frac{1}{2}, \frac{1}{2})$-space and $A$ being of a special form. As a consequence we obtain a new proof for the optimal angle of $\mathrm{L}^{p}$-analyticity for such semigroups, which is essentially the same as in the well-known sub-Markovian case. The proof of the main theorem is a combination of well-known reduction techniques and some representation results about operators on $\mathrm{C}(K)$-spaces. One focus of the paper lies on presenting these auxiliary techniques and results in great detail.

math.FA↗
arXiv2013

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a Hilbert space, then for each $τ>0$ the operator $A$ has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy $f(z)=O(e^{-τ\textrm{Re}(z)})$ as $|z|\rightarrow \infty$. The bound of this calculus grows at most logarithmically as $τ\searrow 0$. As a consequence, $f(A)$ is a bounded operator for each holomorphic function $f$ (on a right half-plane) with polynomial decay at $\infty$. Then we show that each semigroup generator has a so-called (strong) $m$-bounded calculus for all $m\in\mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called $γ$-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.

math.FA↗
arXiv2013

Square Function Estimates and Functional Calculi

In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F, Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f) for certain H-valued functions f on O. The assignment f to Phi_gamma(f) then becomes a vectorial functional calculus, and a "square function estimate" for f simply means the boundedness of Phi_gamma(f). In this view, all results linking square function estimates with the boundedness of a certain (usually the H-infinity) functional calculus simply assert that certain square function estimates imply other square function estimates. In the present paper several results of this type are proved in an abstract setting, based on the principles of subordination, integral representation, and a new boundedness concept for subsets of Hilbert spaces, the so-called ell-1 -frame-boundedness. These abstract results are then applied to the H-infinity calculus for sectorial and strip type operators. For example, it is proved that any strip type operator with bounded scalar H-infinity calculus on a strip over a Banach space with finite cotype has a bounded vectorial H-infinity calculus on every larger strip.

math.FA↗
arXiv2011

Bernstein functions and rates in mean ergodic theorems for operator semigroups

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form $g(A)$, where $-A$ is the generator of the semigroup and $g$ is a Bernstein function. In addition, we obtain some new results on Bernstein functions that are of independent interest.

math.FA↗