Geodesic
Chain-finite operators and locally chain-finite operators
Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 113-121

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We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that TkBTk =Tk and TkB=BTk. Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1). 
Bermúdez, Teresa; Martinón, Antonio. Chain-finite operators and locally chain-finite operators. Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 113-121. doi: 10.1017/S0017089501010096
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     title = {Chain-finite operators and locally chain-finite operators},
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     year = {2001},
     volume = {43},
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     doi = {10.1017/S0017089501010096},
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