Mutually EP operators on Hilbert Spaces: Characterizations and Algebraic Properties

For a closed-range operator on a Hilbert space, the EP property ischaracterized by the equality of the range and the adjoint range.In this work, we introduce a binary relation, called mutual EP,between closed-range operators. We say that two operators$\mathcal{X}, \mathcal{Y} \in \mathcal{B}^\dagger(\mathcal{H})$(i.e., bounded linear operators with closed range) are mutually EP,denoted $\mathcal{X}\,\mathfrak{mEP}\,\mathcal{Y}$(read as ``$\mathcal{X}$ is mutually EP with $\mathcal{Y}$''),whenever$\mathcal{X}^\dagger\mathcal{X}=\mathcal{Y}\mathcal{Y}^\dagger$and$\mathcal{Y}^\dagger\mathcal{Y}=\mathcal{X}\mathcal{X}^\dagger$.The terminology ``mutually EP'' refers to these crossedMoore-Penrose projection identities and does not imply that eitheroperator is individually EP. A key structural finding is that$\mathfrak{mEP}$ is an equivalence relation on$\mathcal{EP}(\mathcal{H})$, the class of closed-range EP operators,but not on all of $\mathcal{B}^\dagger(\mathcal{H})$. Moreover, therelation preserves the partition of$\mathcal{B}^\dagger(\mathcal{H})$ into EP and non-EP operators:$\mathcal{X}\,\mathfrak{mEP}\,\mathcal{Y}$ implies$\mathcal{X} \in \mathcal{EP}(\mathcal{H})\iff\mathcal{Y} \in \mathcal{EP}(\mathcal{H})$.We characterize the relation in terms of range equalities andestablish algebraic properties: invariance under adjoints, unitarytransformations, direct sums, and inverses. Several examplesillustrate the main results. Our results clarify how binary relations between closed-range operatorscan be described through their Moore-Penrose orthogonal projectionsonto the range and adjoint range.