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# On Schauder’s Theorem and $s$-numbers (Daniel Akech Thiong, CGU)

## October 6, 2022 @ 4:00 pm - 5:00 pm

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.

1. 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 Banach
spaces, 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 nests
of subspaces and the nest approximation properties.” Journal of Functional
Analysis 273.9 (2017): 2916-2938.
7. A.Pietsch, Operator ideals, North-Holland, Amsterdam, 1980.

## Details

Date:
October 6, 2022
Time:
4:00 pm - 5:00 pm
Event Category:

Asuman Aksoy