  Uncertainty relations between two general non-commuting self-adjoint
operators are derived in a Krein space. All of these relations involve a Krein
space induced fundamental symmetry operator, $J$, while some of these
generalized relations involve an anti-commutator, a commutator, and various
other nonlinear functions of the two operators in question. As a consequence
there exist classes of non-self-adjoint operators on Hilbert spaces such that
the non-vanishing of their commutator implies an uncertainty relation. All
relations include the classical Heisenberg uncertainty principle as formulated
in Hilbert Space by Von Neumann and others. In addition, we derive an operator
dependent (nonlinear) commutator uncertainty relation in Krein space.
