  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 $T^*$.
