Summary: Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator $\hat J$ such that A is $\hat J$ -Hermitian, that is, $\hat J A = A * \hat J$ , is found. Moreover, we construct a positive definite Hermitian Q such that A is Q-Hermitian, allowing for the standard probabilistic interpretation of quantum mechanics. Finally, it is shown that the considered matrices are similar to Hermitian matrices.

47B50, 47A63

infinite matrices, pseudo-Hermitian matrices, creation and annihilation operators, Kreĭn spaces