Equality in Degrees of Compactness: Schauder's Theorem and s-numbers

Asuman Güven Aksoy; Daniel Akech Thiong

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arXiv · 2306.03629

Equality in Degrees of Compactness: Schauder's Theorem and s-numbers

Asuman Güven Aksoy·Daniel Akech Thiong

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Abstract

We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of $T$ and $T^*$ are equal for compact operators. Second, for non-compact, bounded linear operators from $X$ to $Y$, we obtain a relationship between certain $s$-numbers of $T$ and $T^*$ under natural conditions on $X$ and $Y$. Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results by comparing the degree of compactness of $T$ with that of its adjoint $T^*$.

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BibTeX RIS

Asuman Güven Aksoy, Daniel Akech Thiong. 2023-06-06. Equality in Degrees of Compactness: Schauder's Theorem and s-numbers. https://arxiv.org/abs/2306.03629

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