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.
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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.
References
[a1] | I.C. Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 185–264 Uspekhi Mat. Nauk. , 12 (1957) pp. 43–118 |
[a2] | T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322 |
Semi-Fredholm operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Fredholm_operator&oldid=12472