The dual of a space of compact operators

Keun Young Lee; Gwanghyun Jo; Keun Young Lee; Gwanghyun Jo

doi:10.3934/math.2024473

AIMS Mathematics

2024, Volume 9, Issue 4: 9682-9691. doi: 10.3934/math.2024473

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The dual of a space of compact operators

Keun Young Lee ¹,
Gwanghyun Jo ^{2
,
,}

1.
Independent scholar, Republic of Korea
2.
Department of Mathematical Data Science, Hanyang University ERICA, Ansan, Republic of Korea

Received: 08 December 2023 Revised: 22 February 2024 Accepted: 26 February 2024 Published: 11 March 2024
MSC : 46B20, 46B25, 46B28

Let $ X $ and $ Y $ be Banach spaces. We provide the representation of the dual space of compact operators $ K(X, Y) $ as a subspace of bounded linear operators $ \mathcal{L}(X, Y) $. The main results are: (1) If $ Y $ is separable, then the dual forms of $ K(X, Y) $ can be represented by the integral operator and the elements of $ C[0, 1] $. (2) If $ X^{**} $ has the weak Radon-Nikodym property, then the dual forms of $ K(X, Y) $ can be represented by the trace of some tensor products.
- compact operators,
- Banach space,
- dual space,
- Radon-Nikodym property
Citation: Keun Young Lee, Gwanghyun Jo. The dual of a space of compact operators[J]. AIMS Mathematics, 2024, 9(4): 9682-9691. doi: 10.3934/math.2024473

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Abstract

Let $ X $ and $ Y $ be Banach spaces. We provide the representation of the dual space of compact operators $ K(X, Y) $ as a subspace of bounded linear operators $ \mathcal{L}(X, Y) $. The main results are: (1) If $ Y $ is separable, then the dual forms of $ K(X, Y) $ can be represented by the integral operator and the elements of $ C[0, 1] $. (2) If $ X^{**} $ has the weak Radon-Nikodym property, then the dual forms of $ K(X, Y) $ can be represented by the trace of some tensor products.

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