# Semi-Fredholm operator

Let and be two Banach spaces and let denote the Banach space of all continuous (bounded) operators from into (cf. also Banach space; Continuous operator). For an operator in , let be the set of all such that and let be the quotient space , where denotes the range of . By definition, is a semi-Fredholm operator if is closed (i.e. it is a normally-solvable operator) and at least one of the vector spaces and is of finite dimension. (The definition is partially redundant, since if the dimension of is finite, is closed.)

For a semi-Fredholm operator , its index, i.e. is uniquely determined either as an integer, or as plus or minus infinity. In the first case is a Fredholm operator. Cf. also Index of an operator.

The set of all semi-Fredholm operators in is open in and the index is constant on each connected component of . Moreover, if is a compact operator in and is in , then is also in and its index equals that of . Due to these properties, semi-Fredholm operators play an important role in linear and non-linear analysis. They were first explicitly considered by I.C. Gohberg and M.G. Krein [a1] and T. Kato [a2], who also treated the case when is unbounded.

How to Cite This Entry:
Semi-Fredholm operator. C. Foias (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Fredholm_operator&oldid=12472
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098