Pseudo-Hermiticity, Anti-Pseudo-Hermiticity, and Generalized Parity-Time-Reversal Symmetry at Exceptional Points

For a diagonalizable linear operator $H:\mathscr{H}\to\mathscr{H}$ acting in a separable Hilbert space $\mathscr{H}$, 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 $\mathscr{H}$, 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 $X:\mathscr{H}\to\mathscr{H}$ satisfying $[X,H]=0$ and $X^2=1$. {The original proof of this result makes use of the anti-pesudo-Hermiticity of every diagonalizable operator $L:\mathscr{H}\to\mathscr{H}$, which means the existence of an antilinear Hermitian bijection $\tau:\mathscr{H}\to\mathscr{H}$ satisfying $L^\dagger=\tau L\,\tau^{-1}$. 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 $\mathscr{H}$. {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 $\tau$ and $X$ that realize their anti-pseudo-Hermiticity and generalized $PT$-symmetry at and away from the exceptional points.