Editing Fredholm operator (section)

==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>.