  In recent years the coincidence of the operator relations equivalence after
extension and Schur coupling was settled for the Hilbert space case, by showing
that equivalence after extension implies equivalence after one-sided extension.
In this paper we investigate consequences of equivalence after extension for
compact Banach space operators. We show that generating the same operator ideal
is necessary but not sufficient for two compact operators to be equivalent
after extension. In analogy with the necessary and sufficient conditions on the
singular values for compact Hilbert space operators that are equivalent after
extension, we prove the necessity of similar relationships between the
$s$-numbers of two compact Banach space operators that are equivalent after
extension, for arbitrary $s$-functions.
  We investigate equivalence after extension for operators on
$\ell^{p}$-spaces. We show that two operators that act on different
$\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such
operators can still be equivalent after extension, for instance all invertible
operators are equivalent after extension, however, if one of the two operators
is compact, then they cannot be equivalent after extension. This contrasts the
Hilbert space case where equivalence after one-sided extension and equivalence
after extension are, in fact, identical relations.
  Finally, for general Banach spaces $X$ and $Y$, we investigate consequences
of an operator on $X$ being equivalent after extension to a compact operator on
$Y$. We show that, in this case, a closed finite codimensional subspace of $Y$
must embed into $X$, and that certain general Banach space properties must
transfer from $X$ to $Y$. We also show that no operator on $X$ can be
equivalent after extension to an operator on $Y$, if $X$ and $Y$ are
essentially incomparable Banach spaces.
