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##### CLASSES OF NON-HERMITIAN OPERATORS WITH REAL EIGENVALUES

**Authors**: Bebiano N, da Providencia J, da Providencia JP

**Ref.**: ELECTRONIC JOURNAL OF LINEAR ALGEBRA **21**, 98 (2010)

**Abstract**: 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 (J) over cap such that A is (J) over cap -Hermitian, that is, (J) over cap A = A* (J) over cap, 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.