Mutually EP operators on Hilbert Spaces: Characterizations and Algebraic Properties
Abstract
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.
Keywords
Mutually EP operators, Moore-Penrose inverse, Range projection, Hilbert space operators, Equivalence relation, EP operators
References
- S. L. Campbell, C. D. Meyer, EP operators and generalized inverses, Canad. Math. Bull., 18(3) (1975), 327–333. https://doi.org/10.4153/CMB-1975-061-4
- M. Itoh, On some EP operators, Nihonkai Math. J., 16 (2005), 49–56.
- O. Stanimirovic, D. Mosic, New properties of the core-EP pre-order, Filomat, 39(3) (2025), 743–754. https://doi.org/10.2298/FIL2503743S
- D. Mosic, D. S. Djordjevic, The gDMP inverse of Hilbert space operators, J. Spectr. Theory, 8(2) (2018), 555–573. https://doi.org/10.4171/JST/207
- X. Wang, C. Deng, Properties of $m$-EP operators, Linear Multilinear Algebra, 65(7) (2017), 1349–1361. https://doi.org/10.1080/03081087.2016.1235131
- S. Menkad, A. Elgues, Some results of $n$-EP operators on Hilbert spaces, Int. J. Anal. Appl., 22 (2024), Article ID 78. https://doi.org/10.28924/2291-8639-22-2024-78
- M. M. Karizaki, M. Sohrabi, Applications of polar decomposition of EP operators, Rend. Circ. Mat. Palermo Ser. II, 74(5) (2025), Article ID 143. https://doi.org/10.1007/s12215-025-01262-0
- A. B. Patel, M. P. Shekhawat, Hypo-EP operators, Indian J. Pure Appl. Math., 47(1) (2016), 73–84. https://doi.org/10.1007/s13226-015-0168-x
- A. A. Jibril, On mutually normal operators, Abhath Al-Yarmouk, 8(1) (1999), 53–60.
- P. S. Johnson, Closed EP and hypo-EP operators on Hilbert spaces, J. Anal., 30(4) (2022), 1377–1390. https://doi.org/10.1007/s41478-022-00401-5
