Editing Fredholm operator

{{Short description|Part of Fredholm theories in integral equations}}
{{main|Fredholm theory}}

In [[mathematics]], '''Fredholm operators''' are certain [[Operator (mathematics)|operators]] that arise in the [[Fredholm theory]] of [[integral equation]]s. They are named in honour of [[Erik Ivar Fredholm]]. By definition, a Fredholm operator is a [[bounded linear operator]] ''T''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' between two [[Banach space]]s with finite-dimensional [[kernel (algebra)|kernel]] <math>\ker T</math> and finite-dimensional (algebraic) [[cokernel]] <math>\operatorname{coker}T = Y/\operatorname{ran}T</math>, and with closed [[range of a function|range]] <math>\operatorname{ran}T</math>. The last condition is actually redundant.<ref>{{cite book
| last1=Abramovich | first1=Yuri A.
| last2=Aliprantis | first2=Charalambos D.
| title=An Invitation to Operator Theory
| publisher=American Mathematical Society
| series=Graduate Studies in Mathematics
| volume=50
| date=2002
| isbn=978-0-8218-2146-6
| page=156}}</ref>

The ''[[Linear transform#Index|index]]'' of a Fredholm operator is the integer

:<math> \operatorname{ind}T := \dim \ker T - \operatorname{codim}\operatorname{ran}T </math>

or in other words,

:<math> \operatorname{ind}T := \dim \ker T - \operatorname{dim}\operatorname{coker}T.</math>

==Properties==
Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' between Banach spaces  ''X'' and ''Y'' is Fredholm if and only if it is invertible [[Quotient ring|modulo]] [[compact operator]]s, i.e., if there exists a bounded linear operator

:<math>S: Y\to X</math>

such that

:<math> \mathrm{Id}_X - ST \quad\text{and}\quad \mathrm{Id}_Y - TS </math>

are compact operators on ''X'' and ''Y'' respectively.

If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators from ''X'' to ''Y'' is open in the Banach space L(''X'',&nbsp;''Y'') of bounded linear operators, equipped with the [[operator norm]], and the index is locally constant.  More precisely, if ''T''<sub>0</sub> is Fredholm from ''X'' to ''Y'', there exists ''ε''&nbsp;> 0 such that every ''T'' in L(''X'',&nbsp;''Y'') with {{nowrap begin}}||''T'' &minus; ''T''<sub>0</sub>|| < ''ε''{{nowrap end}} is Fredholm, with the same index as that of&nbsp;''T''<sub>0</sub>.

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

:<math>\operatorname{ind} (U \circ T) = \operatorname{ind}(U) + \operatorname{ind}(T).</math>

When ''T'' is Fredholm, the [[Dual space#Transpose of a continuous linear map|transpose]] (or adjoint) operator {{nowrap|''T''&thinsp;&prime;}} is Fredholm from {{nowrap|''Y''&thinsp;&prime;}} to {{nowrap|''X''&thinsp;&prime;}}, and {{nowrap|ind(''T''&thinsp;&prime;) {{=}} &minus;ind(''T'')}}.  When ''X'' and ''Y'' are [[Hilbert space]]s, the same conclusion holds for the [[Hermitian adjoint]]&nbsp;''T''<sup>∗</sup>.

When ''T'' is Fredholm and ''K'' a compact operator, then ''T''&nbsp;+&nbsp;''K'' is Fredholm.  The index of ''T'' remains unchanged under such a compact perturbations of ''T''.  This follows from the fact that the index ''i''(''s'') of {{nowrap|''T'' + ''s''&thinsp;''K''}} is an integer defined for every ''s'' in [0,&nbsp;1], and ''i''(''s'') is locally constant, hence ''i''(1)&nbsp;=&nbsp;''i''(0).

Invariance by perturbation is true for larger classes than the class of compact operators.  For example, when ''U'' is Fredholm and ''T'' a [[strictly singular operator]], then ''T''&nbsp;+&nbsp;''U'' is Fredholm with the same index.<ref>{{cite journal
| last1=Kato | first1=Tosio
| title=Perturbation theory for the nullity deficiency and other quantities of linear operators
| journal=[[Journal d'Analyse Mathématique]]
| volume=6
| date=1958
| pages=273–322
| doi=10.1007/BF02790238 | doi-access=| s2cid=120480871
}}</ref>  The class of [[strictly singular operator#definitions|inessential operators]], which properly contains the class of strictly singular operators, is the "perturbation class" for Fredholm operators.  This means an operator <math>T\in B(X,Y)</math> is inessential if and only if ''T+U'' is Fredholm for every Fredholm operator <math>U\in B(X,Y)</math>.

==Examples==
Let <math>H</math> be a [[Hilbert space]] with an orthonormal basis <math>\{e_n\}</math> indexed by the non negative integers.  The (right) [[shift operator]] ''S'' on ''H'' is defined by

:<math>S(e_n) = e_{n+1}, \quad n \ge 0. \,</math>

This operator ''S'' is injective (actually, isometric) and has a closed range of codimension 1, hence ''S'' is Fredholm with <math>\operatorname{ind}(S)=-1</math>.  The powers <math>S^k</math>, <math>k\geq0</math>, are Fredholm with index <math>-k</math>.  The adjoint ''S*'' is the left shift,

:<math>S^*(e_0) = 0, \ \ S^*(e_n) = e_{n-1}, \quad n \ge 1. \,</math>

The left shift ''S*'' is Fredholm with index&nbsp;1.

If ''H'' is the classical [[Hardy space]] <math>H^2(\mathbf{T})</math> on the unit circle '''T''' in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

:<math>e_n : \mathrm{e}^{\mathrm{i} t} \in \mathbf{T} \mapsto
\mathrm{e}^{\mathrm{i} n t}, \quad n \ge 0, \, </math>

is the multiplication operator ''M''<sub>''φ''</sub> with the function <math>\varphi=e_1</math>.  More generally, let ''φ'' be a complex continuous function on '''T''' that does not vanish on <math>\mathbf{T}</math>, and let ''T''<sub>''φ''</sub> denote the [[Toeplitz operator]] with symbol ''φ'', equal to multiplication by ''φ'' followed by the orthogonal projection <math>P:L^2(\mathbf{T})\to H^2(\mathbf{T})</math>:

:<math> T_\varphi : f \in H^2(\mathrm{T}) \mapsto P(f \varphi) \in H^2(\mathrm{T}). \, </math>

Then ''T''<sub>''φ''</sub> is a Fredholm operator on <math>H^2(\mathbf{T})</math>, with index related to the [[winding number]] around 0 of the closed path <math>t\in[0,2\pi]\mapsto \varphi(e^{it})</math>:  the index of  ''T''<sub>''φ''</sub>, as defined in this article, is the opposite of this winding number.

==Applications==
Any [[elliptic operator]] on a closed manifold 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.

The [[Atiyah-Singer index theorem]] gives a topological characterization of the index of certain operators on manifolds.

The [[Atiyah-Jänich theorem]] identifies the [[Topological K-theory|K-theory]] ''K''(''X'') of a compact topological space ''X'' with the set of [[homotopy class]]es of continuous maps from ''X'' to the space of Fredholm operators ''H''&rarr;''H'', where ''H'' is the separable Hilbert space and the set of these operators carries the operator norm.

== Generalizations ==

=== Semi-Fredholm operators ===
A bounded linear operator ''T'' is called '''semi-Fredholm''' if its range is closed and at least one of <math>\ker T</math>, <math>\operatorname{coker}T</math> is finite-dimensional. For a semi-Fredholm operator, the index is defined by

:<math>
\operatorname{ind}T=\begin{cases}
+\infty,&\dim\ker T=\infty;
\\
\dim\ker T-\dim\operatorname{coker}T,&\dim\ker T+\dim\operatorname{coker}T<\infty;
\\
-\infty,&\dim\operatorname{coker}T=\infty.
\end{cases}
</math>

===Unbounded operators===
One may also define unbounded Fredholm operators. Let ''X'' and ''Y'' be two Banach spaces.

# The [[Unbounded_operator#Closed_linear_operators|closed linear operator]] <math>T:\,X\to Y</math> is called ''Fredholm'' if its domain <math>\mathfrak{D}(T)</math> is dense in <math>X</math>, its range is closed, and both kernel and cokernel of ''T'' are finite-dimensional.
#<math>T:\,X\to Y</math> is called ''semi-Fredholm'' if its domain <math>\mathfrak{D}(T)</math> is dense in <math>X</math>, its range is closed, and either kernel or cokernel of ''T'' (or both) is finite-dimensional.

As it was noted above, the range of a closed operator is closed as long as the cokernel is finite-dimensional (Edmunds and Evans, Theorem I.3.2).

==Notes==
{{wikibooks
 |1= Functional Analysis
 |2= Special topics
 |3= Fredholm theory
}}
<references/>

==References==
* D.E. Edmunds and W.D. Evans (1987), ''Spectral theory and differential operators,'' Oxford University Press. {{ISBN|0-19-853542-2}}.
* A. G. Ramm, "[http://www.math.ksu.edu/~ramm/papers/419.pdf A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators]", ''American Mathematical Monthly'', '''108''' (2001) p.&nbsp;855 (NB: In this paper the word "Fredholm operator" refers to "Fredholm operator of index 0").
* {{mathworld|urlname=FredholmsTheorem|title=Fredholm's Theorem}}
* {{springer|id=f/f041470|title=Fredholm theorems|author=B.V. Khvedelidze}}
* 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.&nbsp;579–600.
* 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.
* 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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[[Category:Fredholm theory]]
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