  We introduce the notion of C-orbit reflexivity and study its properties. An
operator on a finite-dimensional space is C-orbit reflexive if and only if the
two largest blocks in its Jordan form corresponding to nonzero eigenvalues with
the largest modulus differ in size by at most one. Most of the proofs of our
results in infinite dimensions are obtained from purely algebraic results we
obtain from linear-algebraic analogs of C-orbit reflexivity.
