OICC Press Mathematical Analysis and its Contemporary Applications 7 2 2025 05 26 Orthogonality of algebraic elementary operators when their numerical ranges are spheroidal 10.30495/maca.2025.2055200.1132 EN James Mwangi Njoroge Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, P.O. Box 210-40601, Bondo, Kenya Benard Okelo Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, P.O. Box 210-40601, Bondo, Kenya https://orcid.org/0000-0003-3963-1910 Priscah Omoke Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, P.O. Box 210-40601, Bondo, Kenya Journal Article 2025 05 26 Characterizations involving algebraic elementary operators have been done over the years, for instance, orthogonality when the operators are induced by other different types of transformations. In particular, algebraic elementary operators induced by norm-attainable maps have not been characterized in terms of orthogonality when their numerical ranges have spheroid boundaries. In this note, we characterize algebraic elementary operators in terms of Birkhoff-James orthogonality when they are induced by norm-attainable maps and the boundaries of their numerical ranges are spheroidal in shape. We show that under the spheroidicity criterion for the numerical range boundary, various types of algebraic elementary operators satisfy Birkhoff-James orthogonality.