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 operatorsX, Y ∈ B⁽H)(i.e., bounded linear operators with closed range) are mutually EP,denoted X\,mEP\,Y(read as ``X is mutually EP with Y''),wheneverX⁼YY^andY⁼XX^.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 thatmEP is an equivalence relation onEP(H), the class of closed-range EP operators,but not on all of B⁽H). Moreover, therelation preserves the partition ofB⁽H) into EP and non-EP operators:X\,mEP\,Y impliesX ∈ EP(H) ∈ EP(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.