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Most recently it has been observed e.g. by Bender and Klevansky(arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetricnon-Hermitian Hamilton operator is not unique. Moreover it has been remarked byShi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exista well defined inner product in the context of the Hamilton operator of thePT-symmetric non-Hermitian Lee model yielding a different C-operator ascompared to the one previously derived by Bender et al.. The puzzlingobservations of both manuscripts are reconciled and explained in the presentmanuscript as follows: the actual form of the metric operator (and the inducedC-operator) related to some non-Hermitian Hamilton operator constructed alongthe lines of Shi and Su depends on the chosen normalization of the left andright eigenvectors of the Hamilton operator under consideration and istherefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operatorit is shown that - by a suitable choice of the norm of its eigenvectors - themetric operator yielding a positive semi-definite inner product can be madeeven independent of the parameters of the considered Hamilton operator. Thissurprising feature makes in turn the obtained metric operator rather unique andattractive. For later convenience the metric operator for the Bosonic andFermionic (anti)causal harmonic oscillator is derived.

The construction of a general inner product in non-Hermitian quantum  theory and some explanation for the nonuniqueness of the C operator in PT  quantum mechanics