  In the recent years a generalization of Hermiticity was investigated using a
complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator
Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians,
possessing PT symmetry (the product of parity and time reversal), can have real
spectrum. We will consider the most simple case: \epsilon even. In this paper
we describe all self-adjoint (Hermitian) and at the same time PT symmetric
operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that
there are a large class of self-adjoint operators associated to H=p^2
+x^2(ix)^\epsilon which are not PT symmetric.
