Majumdar, A.Johnson, P.S.2026-02-032025Linear and Multilinear Algebra, 2025, 73, 8, pp. 1668-16843081087https://doi.org/10.1080/03081087.2024.2436091https://idr.nitk.ac.in/handle/123456789/20515In this paper, we present some interesting results to characterize the Moore–Penrose inverses of unbounded closable operators and after investigating the Moore–Penrose inverses of the direct sum of closed operators with closed ranges in Hilbert spaces, we establish (Formula presented.), where (Formula presented.) is the reduced minimum modulus of T. © 2024 Informa UK Limited, trading as Taylor & Francis Group.closable operatorclosed operatorMoore–Penrose inverse