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

We investigate an extension of Schauder’s theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^{\ast }$. We have three main results. First, we present a new proof that the approximation number of $T$ and $T^{\ast }$ 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 $T^{\ast }$ 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 $T^{\ast }$.