  In our previous work, we proposed a mathematical framework for PT-symmetric
quantum theory, and in particular constructed a Krein space in which
PT-symmetric operators would naturally act. In this work, we explore and
discuss various general consequences and aspects of the theory defined in the
Krein space, not only spectral property and PT symmetry breaking but also
several issues, crucial for the theory to be physically acceptable, such as
time evolution of state vectors, probability interpretation, uncertainty
relation, classical-quantum correspondence, completeness, existence of a basis,
and so on. In particular, we show that for a given real classical system we can
always construct the corresponding PT-symmetric quantum system, which indicates
that PT-symmetric theory in the Krein space is another quantization scheme
rather than a generalization of the traditional Hermitian one in the Hilbert
space. We propose a postulate for an operator to be a physical observable in
the framework.
