  It is proved that the resolvent norm of an operator with a compact resolvent
on a Banach space $X$ cannot be constant on an open set if the underlying space
or its dual is complex strictly convex. It is also shown that this is not the
case for an arbitrary Banach space: there exists a separable, reflexive space
$X$ and an unbounded, densely defined operator acting in $X$ with a compact
resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is
isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a
bounded linear operator acting on the same space whose resolvent norm is
constant in a neighbourhood of zero. It is shown that similar examples cannot
exist in the co-dimension $1$ case.
