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DTSTART;TZID=America/Los_Angeles:20221006T160000
DTEND;TZID=America/Los_Angeles:20221006T170000
DTSTAMP:20260501T041006
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:20220922T160000
DTEND;TZID=America/Los_Angeles:20220922T170000
DTSTAMP:20260501T041006
CREATED:20220918T041430Z
LAST-MODIFIED:20220918T041430Z
UID:2930-1663862400-1663866000@colleges.claremont.edu
SUMMARY:Frobenius-Rieffel norms on matrix algebras (Konrad Aguilar\, Pomona)
DESCRIPTION:Noncommutative metric geometry is the study of certain noncommuative algebras in the context of metric geometry. For instance\, the Lipschitz constant (which measures the maximum slope obtained by a real-valued continuous function on a metric space (allowed to be infinite)) is a vital tool in metric geometry\, and a main feature of noncommutative metric geometry is the introduction of a noncommutative notion of the Lipschitz constant\, called an L-seminorm\, due to M.A. Rieffel. The purpose of our work is to introduce suitable L-seminorms on matrix algebras. To accomplish this\, we used norms introduced by Rieffel on certain unital C*-algebras built from conditional expectations onto unital C*-subalgebras. We begin by showing that these norms generalize the Frobenius norm on matrix algebras\, and we provide explicit formulas for certain conditional expectations onto unital C*-subalgebras of finite-dimensional C*-algebras. This allows us to compare these norms to the unique C*-norm (the operator 2-norm)\, by finding explicit equivalence constants. (This is joint work with Stephan R. Garcia and Elena Kim (’21)\, arxiv: 2112.13164).
URL:https://colleges.claremont.edu/ccms/event/frobenius-rieffel-norms-on-matrix-algebras-konrad-aguilar-pomona/
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
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20220908T160000
DTEND;TZID=America/Los_Angeles:20220908T170000
DTSTAMP:20260501T041006
CREATED:20220905T060933Z
LAST-MODIFIED:20230816T041748Z
UID:2824-1662652800-1662656400@colleges.claremont.edu
SUMMARY:Factorization theorems of Backward Shifts and Nuclear Maps (Asuman Aksoy\, CMC)
DESCRIPTION:The theory of compact linear operators between Banach spaces has a classical core and is familiar to many. Perhaps lesser known is the factorization of compact maps through a closed subspace of \(c_0\) [2]. This factorization theorem has a number of important connections and consequences analogous to how the ideals of continuous linear operators factoring compactly through \(\ell^p\)-spaces \((1\leq p < \infty)\) (see [1] and the references therein). In this talk\, even though hypercyclic operators are not compact\, we consider operator ideals generated by hypercyclic backward weighted shifts and examine their factorization properties. (Joint work with Yunied Puig)\n\n\n\nFourie\, Jan H. Injective and surjective hulls of classical \(p\)-compact operators with application to unconditionally \(p\)-compact operators. Studia Math.  240  (2018)\, no. 2\, 147–159. MR3720927\nTerzioğlu\, T. A characterization of compact linear mappings. Arch. Math. (Basel) 22 (1971)\, 76–78. MR0291865
URL:https://colleges.claremont.edu/ccms/event/factorization-theorems-of-backward-shifts-and-nuclear-maps-asuman-aksoy-cmc/
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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