Convergence of operators with closed range

Abstract

Izumino has discussed a sequence of closed range operators (T<inf>n</inf>) that converges to a closed range operator T on a Hilbert space to establish the convergence of T<inf>n</inf>† ? T† for Moore-Penrose inverses. In general, if Tn ? T uniformly and each Tn has a closed range, then T need not have a closed range. Some sufficient conditions have been discussed on T<inf>n</inf> and T such that T has a closed range whenever each T<inf>n</inf> has a closed range. © 2019 Khayyam Journal of Mathematics.