# Convergence of Operators with Closed Range

Document Type : Original Article

Authors

1 Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Karnataka - 575 025, India.

2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamilnadu - 632 014, India.

Abstract

Izumino has discussed a sequence of closed range operators $(T_n)$ that converges to a closed range operator $T$ on a Hilbert space to establish the convergence of $T^{\dag}_n$ $\to$ $T^{\dag}$ for Moore-Penrose inverses. In general, if $T_n \to T$ uniformly and each $T_n$ has a closed range, then $T$ need not have a closed range. Some sufficient conditions have been discussed on $T_n$ and $T$ such that $T$ has a closed range whenever each $T_n$ has a closed range.

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