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| In [[mathematics]], a '''Fredholm operator''' is an [[Operator (mathematics)|operator]] that arises in the [[Fredholm theory]] of [[integral equation]]s. It is named in honour of [[Erik Ivar Fredholm]].
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| A Fredholm operator is a [[bounded linear operator]] between two [[Banach space]]s whose [[kernel (algebra)|kernel]] and [[cokernel]] are finite-dimensional and whose [[range (mathematics)|range]] is closed. (The last condition is actually redundant.<ref>Yuri A. Abramovich and Charalambos D. Aliprantis, "An Invitation to Operator Theory", p.156</ref>) Equivalently, an operator ''T'' : ''X'' → ''Y'' is Fredholm if it is invertible [[Quotient_ring|modulo]] [[compact operator]]s, i.e., if there exists a bounded linear operator
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| :<math>S: Y\to X</math>
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| such that
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| :<math> \mathrm{Id}_X - ST \quad\text{and}\quad \mathrm{Id}_Y - TS </math>
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| are compact operators on ''X'' and ''Y'' respectively.
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| The ''index'' of a Fredholm operator is
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| :<math> \mathrm{ind}\,T := \dim \ker T - \mathrm{codim}\,\mathrm{ran}\,T </math>
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| or equivalently,
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| :<math> \mathrm{ind}\,T := \dim \ker T - \mathrm{dim}\,\mathrm{coker}\,T;</math>
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| see [[dimension]], [[null space|kernel]], [[codimension]], range, and cokernel.
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| ==Properties==
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| The set of Fredholm operators from ''X'' to ''Y'' is open in the Banach space L(''X'', ''Y'') of bounded linear operators, equipped with the [[operator norm]]. More precisely, when ''T''<sub>0</sub> is Fredholm from ''X'' to ''Y'', there exists ''ε'' > 0 such that every ''T'' in L(''X'', ''Y'') with {{nowrap begin}}||''T'' − ''T''<sub>0</sub>|| < ''ε''{{nowrap end}} is Fredholm, with the same [[Linear_transform#Index|index]] as that of ''T''<sub>0</sub>.
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| When ''T'' is Fredholm from ''X'' to ''Y'' and ''U'' Fredholm from ''Y'' to ''Z'', then the composition <math>U \circ T</math> is Fredholm from ''X'' to ''Z'' and
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| :<math>\mathrm{ind} (U \circ T) = \mathrm{ind}(U) + \mathrm{ind}(T).</math>
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| When ''T'' is Fredholm, the [[Dual space#Transpose of a continuous linear map|transpose]] (or adjoint) operator {{nowrap|''T'' ′}} is Fredholm from {{nowrap|''Y'' ′}} to {{nowrap|''X'' ′}}, and {{nowrap|ind(''T'' ′) {{=}} −ind(''T'')}}. When ''X'' and ''Y'' are [[Hilbert Space | Hilbert spaces]], the same conclusion holds for the [[Hermitian adjoint]] ''T''<sup>∗</sup>.
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| When ''T'' is Fredholm and ''K'' a compact operator, then ''T'' + ''K'' is Fredholm. The index of ''T'' remains constant under compact perturbations of ''T''. This follows from the fact that the index ''i''(''s'') of {{nowrap|''T'' + ''s'' ''K''}} is an integer defined for every ''s'' in [0, 1], and ''i''(''s'') is locally constant, hence ''i''(1) = ''i''(0).
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| Invariance by perturbation is true for larger classes than the class of compact operators. For example, when ''T'' is Fredholm and ''S'' a strictly singular operator, then ''T'' + ''S'' is Fredholm with the same index.<ref>T. Kato, "Perturbation theory for the nullity deficiency and other quantities of linear operators", ''J. d'Analyse Math''. '''6''' (1958), 273–322.</ref> A bounded linear operator ''S'' from ''X'' to ''Y'' is '''strictly singular''' when its restriction to any infinite dimensional subspace ''X''<sub>0</sub> of ''X'' fails to be an into isomorphism, that is:
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| :<math>\inf \{ \|S x\| : x \in X_0, \, \|x\| = 1 \} = 0. \,</math>
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| ==Examples==
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| Let ''H'' be a [[Hilbert space]] with an orthonormal basis {''e''<sub>''n''</sub>} indexed by the non negative integers. The (right) [[shift operator]] ''S'' on ''H'' is defined by
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| :<math>S(e_n) = e_{n+1}, \quad n \ge 0. \,</math>
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| This operator ''S'' is injective (actually, isometric) and has a closed range of codimension 1, hence ''S'' is Fredholm with ind(''S'') = −1. The powers ''S''<sup>''k''</sup>, ''k'' ≥ 0, are Fredholm with index −''k''. The adjoint ''S''<sup>∗</sup> is the left shift,
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| :<math>S^*(e_0) = 0, \ \ S^*(e_n) = e_{n-1}, \quad n \ge 1. \,</math>
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| The left shift ''S''<sup>∗</sup> is Fredholm with index 1.
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| If ''H'' is the classical [[Hardy space]] ''H''<sup>2</sup>('''T''') on the unit circle '''T''' in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials
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| :<math>e_n : \mathrm{e}^{\mathrm{i} t} \in \mathbf{T} \rightarrow
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| \mathrm{e}^{\mathrm{i} n t}, \quad n \ge 0, \, </math>
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| is the multiplication operator ''M''<sub>''φ''</sub> with the function ''φ'' = ''e''<sub>1</sub>. More generally, let ''φ'' be a complex continuous function on '''T''' that does not vanish on '''T''', and let ''T''<sub>''φ''</sub> denote the [[Toeplitz operator]] with symbol ''φ'', equal to multiplication by ''φ'' followed by the orthogonal projection ''P'' from ''L''<sup>2</sup>('''T''') onto ''H''<sup>2</sup>('''T'''):
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| :<math> T_\varphi : f \in H^2(\mathrm{T}) \rightarrow P(f \varphi) \in H^2(\mathrm{T}). \, </math>
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| Then ''T''<sub>''φ''</sub> is a Fredholm operator on ''H''<sup>2</sup>('''T'''), with index related to the [[winding number]] around 0 of the closed path {{nowrap|''t'' ∈ [0, 2 ''π''] → ''φ''(e<sup> i ''t'' </sup>) }}''':''' the index of ''T''<sub>''φ''</sub>, as defined in this article, is the opposite of this winding number.
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| ==Applications==
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| The [[Atiyah-Singer index theorem]] gives a topological characterization of the index of certain operators on manifolds.
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| An [[elliptic operator]] can be extended to a Fredholm operator. The use of Fredholm operators in [[partial differential equation]]s is an abstract form of the [[parametrix]] method.
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| ==B-Fredholm operators==
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| For each integer <math>n</math>, define <math> T_{n} </math> to be the restriction of <math>T</math> to
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| <math> R(T^{n}) </math> viewed as a map from
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| <math> R(T^{n}) </math> into <math> R(T^{n}) </math> ( in particular <math> T_{0} = T</math>).
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| If for some integer <math>n</math> the space <math> R(T^{n}) </math> is closed and <math> T_{n} </math> is a Fredholm operator,then <math>T </math> is called a B-Fredholm operator. The index of a B-Fredholm operator <math>T</math> is defined as the index of the Fredholm operator <math> T_n </math>. It is shown that the index is independent of the integer <math> n</math>.
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| B-Fredholm operators were introduced by M. Berkani in 1999 as a generalization of Fredholm operators. <ref>Berkani Mohammed: On a class of quasi-Fredholm operators
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| INTEGRAL EQUATIONS AND OPERATOR THEORY
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| Volume 34, Number 2 (1999), 244-249 [http://www.springerlink.com/content/xr3637434785m705/]</ref>
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| ==Notes==
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| {{wikibooks
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| |1= Functional Analysis
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| |2= Special topics
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| |3= Fredholm theory
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| }}
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| <references/>
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| ==References==
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| * D.E. Edmunds and W.D. Evans (1987), ''Spectral theory and differential operators,'' Oxford University Press. ISBN 0-19-853542-2.
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| * A. G. Ramm, "[http://www.math.ksu.edu/~ramm/papers/419amm.pdf A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators]", ''American Mathematical Monthly'', '''108''' (2001) p. 855 (NB: In this paper the word "Fredholm operator" refers to "Fredholm operator of index 0").
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| * {{planetmath_reference|id=3353|title=Fredholm operator}}
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| * {{mathworld|urlname=FredholmsTheorem|title=Fredholm's Theorem}}
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| * {{springer|id=f/f041470|title=Fredholm theorems|author=B.V. Khvedelidze}}
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| * Bruce K. Driver, "[http://math.ucsd.edu/~driver/231-02-03/Lecture_Notes/compact.pdf Compact and Fredholm Operators and the Spectral Theorem]", ''Analysis Tools with Applications'', Chapter 35, pp. 579–600.
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| * Robert C. McOwen, "[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102780323 Fredholm theory of partial differential equations on complete Riemannian manifolds]", ''Pacific J. Math.'' '''87''', no. 1 (1980), 169–185.
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| * Tomasz Mrowka, [http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004/lecture-notes/lecture16_17.pdf A Brief Introduction to Linear Analysis: Fredholm Operators], Geometry of Manifolds, Fall 2004 (Massachusetts Institute of Technology: MIT OpenCouseWare)
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Fredholm Operator}}
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| [[Category:Fredholm theory]]
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