Geodesic
Regularizers of Closed Operators
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 67-71

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Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement. 
Lin, C.-S. Regularizers of Closed Operators. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 67-71. doi: 10.4153/CMB-1974-011-7
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     title = {Regularizers of {Closed} {Operators}},
     journal = {Canadian mathematical bulletin},
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     year = {1974},
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     number = {1},
     doi = {10.4153/CMB-1974-011-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-011-7/}
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