Geodesic
Conditions for the range of a~difference operator to be closed
Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 593-618

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In this paper we study difference operators with variable differences that lie on a straight line in the space of holomorphic functions of a single variable. We obtain necessary and sufficient conditions for the range of such operators to be closed in terms of the zeros of their extreme coefficients. For a wide class of regions these conditions give simple criteria for the range of a difference operator to be closed. 
@article{IM2_1997_61_3_a4,
     author = {S. G. Merzlyakov},
     title = {Conditions for the range of a~difference operator to be closed},
     journal = {Izvestiya. Mathematics },
     pages = {593--618},
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     number = {3},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a4/}
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S. G. Merzlyakov. Conditions for the range of a~difference operator to be closed. Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 593-618. http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a4/