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DTSTART;TZID=America/Los_Angeles:20221006T160000
DTEND;TZID=America/Los_Angeles:20221006T170000
DTSTAMP:20260408T215854
CREATED:20221002T165522Z
LAST-MODIFIED:20230816T042942Z
UID:2946-1665072000-1665075600@colleges.claremont.edu
SUMMARY:On Schauder's Theorem and $s$-numbers (Daniel Akech Thiong\, CGU)
DESCRIPTION:Let \(\mathcal{L}(X\,Y)\) denote the normed vector space of all continuous operators from \(X\) to \(Y\)\, \(X^*\) be the dual space of \(X\)\, and \(\mathcal{K}(X\,Y)\) denote the collection of all compact operators from \(X\) to \(Y\). Denote by \(T^{*} \in \mathcal{L}(Y^{*}\, X^{*} )\) the adjoint operator of \(T\in \mathcal{L} (X\, Y)\). The well known theorem of Schauder states that \(T \in \mathcal{K}(X\,Y) \iff T^{*} \in \mathcal{K}(Y^{*}\,X^{*})\). When an operator fails to be compact\, it is sometimes useful to be able to quantify the degree to which it fails to be compact\, which has led to the introduction of certain approximation quantities\, usually called \(s\)-numbers\, and are closely related to singular values. Specifically\, the concept of \(s\)-numbers\, \(s_n(T)\)\, arises from the need to assign to every operator \(T: X \to Y\) a certain sequence of numbers \(\{s_n(T)\}\) such that \[s_1(T) \geq s_2(T) \geq \dots \geq 0\] which characterizes the degree of compactness/non-compactness of \(T\). The main examples of \(s\)-numbers include approximation numbers and Kolmogorov numbers. Motivated by Schauder’s theorem\, in this talk I will present the relationship between various \(s\)-numbers of an operator \(T\) and its adjoint \(T^*\) between Banach spaces. Joint work with Asuman G. Aksoy. \n1. A. G. Aksoy\, On a theorem of Terzioğlu\, Turk J Math\, 43\, (2019)\, 258-267.2. A. G. Aksoy and M. Nakamura\, The approximation numbers \(\gamma_n(T)\) and Q–compactness\, Math. Japon. 31 (1986)\, no. 6\, 827-840.3. K. Astala\, On measures of non-compactness and ideal variations in Banachspaces\, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertations 29\, (1980)\, 1-42.4. B. Carl and I. Stephani\, Entropy\, compactness and the approximation of oper-ators\, Cambridge University Press\, 1990.5. C. V. Hutton\, On approximation numbers and its adjoint. Math. Ann. 210(1974)\, 277-280.6. Oja\, Eve\, and Silja Veidenberg. ”Principle of local reflexivity respecting nestsof subspaces and the nest approximation properties.” Journal of FunctionalAnalysis 273.9 (2017): 2916-2938.7. A.Pietsch\, Operator ideals\, North-Holland\, Amsterdam\, 1980.
URL:https://colleges.claremont.edu/ccms/event/on-schauders-theorem-and-s-numbers-daniel-akech-thiong-cgu/
LOCATION:Roberts North 105\, CMC\, 320 E. 9th St.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Analysis Seminar
ORGANIZER;CN="Asuman Aksoy":MAILTO:asuman.aksoy@claremontmckenna.edu
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DTSTART;TZID=America/Los_Angeles:20221013T160000
DTEND;TZID=America/Los_Angeles:20221013T170000
DTSTAMP:20260408T215854
CREATED:20221010T130525Z
LAST-MODIFIED:20221010T130525Z
UID:2956-1665676800-1665680400@colleges.claremont.edu
SUMMARY:Quantum metrics on the natural numbers (Katrine von Bornemann Hjelmborg\, University of Southern Denmark)
DESCRIPTION:Quantum metrics in the sense of Rieffel were introduced to prove some statements arising in the high-energy physics literature. Since then\, the area of quantum metric geometry has been used to answer questions stemming from within mathematics as well. To prove such results\, it is often the case that certain properties of a quantum metric are sufficient enough\, and explicit calculations of the quantum metric are rare. Thus\, in this talk\, we focus on certain quantum metrics introduced by Aguilar and Latrémolière on $c$\, the space of complex-valued convergent sequences (which is isomorphic to the space of complex-valued continuous functions on the Alexandroff compactification of the natural numbers)\, and calculate exactly the metrics on the natural numbers that these quantum metrics induce. Moreover\, we compare the quantum metrics of Aguilar and Latrémolière with a classical quantum metric on $c$ induced by the Lipschitz seminorm. (This is joint work with Konrad Aguilar).
URL:https://colleges.claremont.edu/ccms/event/quantum-metrics-on-the-natural-numbers-katrine-von-bornemann-hjelmborg-university-of-southern-denmark/
LOCATION:Roberts North 105\, CMC\, 320 E. 9th St.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Analysis Seminar
ORGANIZER;CN="Asuman Aksoy":MAILTO:asuman.aksoy@claremontmckenna.edu
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