Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1127-1139

Online first article October 16, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

Equality in degrees of compactness: Schauder's theorem and $s$-numbers

Asuman Guven Aksoy, Daniel Akech Thiong

Claremont McKenna College; Claremont Graduate University


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 for comparing the degree of compactness of $T$ with that of its adjoint $T^*$.

Keywords: $s$-numbers, approximation schemes, Schauder's theorem

MSC numbers: Primary 47B06, 47B10; Secondary 47B07

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