Pseudo-hermiticity, anti-pseudo-hermiticity, and generalized parity-time-reversal symmetry at exceptional points

For a diagonalizable linear operator H:H→H acting in a separable Hilbert space H, i.e., an operator with a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of eigenvectors that form a Reisz basis of H, the pseudo-Hermiticity of H is equivalent to its generalized parity-time-reversal (PT) symmetry, where the latter means the existence of an antilinear operator X:H→H satisfying [X,H]=0 and X2=1. The original proof of this result makes use of the anti-pesudo-Hermiticity of every diagonalizable operator L:H→H, which means the existence of an antilinear Hermitian bijection τ:H→H satisfying L† = τL τ−1. We establish the validity of this result for block-diagonalizable operators, i.e., those which have a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of generalized eigenvectors that form a Jordan Reisz basis of H. This allows us to generalize the original proof of the equivalence of pseudo-Hermiticity and generalized PT-symmetry for diagonalizable operators to block-diagonalizable operators. For a pair of pseudo-Hermitian operators acting respectively in two-dimensional and infinite-dimensional Hilbert spaces, we obtain explicit expressions for the antlinear operators τ and X that realize their anti-pseudo-Hermiticity and generalized PT-symmetry at and away from the exceptional points.