On a weak topology for Hadamard spaces

We investigate whether existing notions of weak sequential convergence on Hadamard spaces can be topologized, that is whether there exist corresponding notions of weak topologies. We provide an affirmative answer on what we call weakly proper Hadamard spaces. A notion of dual space is proposed and it is shown that our weak topology and dual space coincide with the standard ones in the case of a Hilbert space. We extend several results from classical functional analysis to the setting of Hadamard spaces, and we compare our topology to existing notions of weak topologies

Investigations in Hadamard spaces

Kjo tezë e doktoratës hulumton ndërveprimin midis gjeometrisë dhe analizës konvekse në hapësirat Hadamard. E motivuar nga aplikime të shumta të gjeometrisë CAT(0), puna jonë bazohet në rezultatet e shumë autorëve të mëparshëmnë mbi analizën konvekse dhe gjeometrinë në sensin e Alexandrovit. Hetimet tona u përgjigjen disa pyetjeve në teorinë e hapësirave CAT(0) prej të cilave disa janë parashtruar si probleme të hapura në literaturën e fundit. Teza jonë e doktoratës zhvillohet sipas linjave të mëposhtme: 1. Topologjitë e dobëta në hapësirat Hadamard, 2. Konveksifikimi i bashkësive kompakte, 3. Problemi i pemës mesatare në hapësirat e pemëve filogjenetike, 4. Konvergjenca Mosko në hapësirat Hadamard, 5. Operatorët (plotësisht) jo-ekspansivë dhe aplikimet e tyre në hapësirat Hadamard.Diese Doktorarbeit untersucht das Zusammenspiel zwischen Geometrie und konvexer Analyse in Hadamardräumen. Motiviert durch zahlreiche Anwendungen der CAT(0)-Geometrie baut unsere Arbeit auf den Ergebnissen vieler früherer Autoren in der konvexen Analysis und der Alexandrov-Geometrie auf. Unsere Untersuchungen beantworten mehrere Fragen in der Theorie von CAT(0)-Räumen, von denen einige in der neueren Literatur als offene Probleme gestellt wurden. Zusammengefasst entwickelt sich unsere Dissertation in folgende Richtungen: 1. Schwache Topologien in Hadamard-Räumen, 2. Konvexe Hüllen kompakter Mengen, 3. Mittleres Baumproblem in phylogenetischen Baumräumen, 4. Mosco-Konvergenz in Hadamard-Räumen, 5. Fest nichtexpansive Operatoren und ihre Anwendungen in Hadamard-Räumen.This thesis investigates the interplay between geometry and convex analysis in Hadamard spaces. Motivated by numerous applications of CAT(0) geometry, our work builds upon the results in convex analysis and Alexandrov geometry of many previous authors. Our investigations answer several questions in the theory of CAT(0) spaces some of which were posed as open problems in recent literature. In a nutshell our thesis develops along the following lines: 1. Weak topologies in Hadamard spaces, 2. Convex hulls of compact sets, 3. Mean tree problem in phylogenetic tree spaces, 4. Mosco convergence in Hadamard spaces, 5. Firmly nonexpansive operators and their applications in Hadamard spaces

Banach spaces of sequences arising from infinite matrices

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences $(\ell_M^p,\|\cdot\|_{M,p})$. In particular we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices $M$ for all $p>1$. A special attention is given to the identification of the dual space $(\ell_M^p )^*$. Building on the earlier works of Bennett and J\"agers, we extend and apply some classical factorization results to the sequence spaces $\ell_M^p$

Duopoly price competition with limited capacity

We study a variation of the duopoly model by Kreps and Scheinkman (1983). Firms limited by their capacity of production engage in a two stage game. In the first stage they commit to levels of production not exceeding their capacities which are then made common knowledge. In the second stage after production has taken place firms simultane- ously compete in prices. Solution of this sequential game shows that the unique Cournot equilibrium outcome as in Kreps and Scheinkman is not always guaranteed. However the Cournot outcome is still robust in the sense that given sufficiently large capacities this equilibrium holds. If capacities are sufficiently small, firms decide to produce at their full capacity and set a price which clears the market at the given level of output.TU Berlin, Open-Access-Mittel – 202

On -Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

We introduce the class of -firmly nonexpansive and quasi -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where -firmly nonexpansive operators coincide with so-called -averaged operators. For our more general setting, we show that -averaged operators form a subset of -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) -firmly nonexpansive. Moreover, we will see that quasi -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates +1:= belong to the fixed point set Fix whenever the operator T is nonexpansive and quasi -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in Fix. Further, the projections Fix converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in , ∈(1,∞)∖{2} spaces on probability measure spaces.DFG, 390685689, EXC 2046: MATH+: Berlin Mathematics Research CenterTU Berlin, Open-Access-Mittel – 202