View source for Semi-Fredholm operator

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302202.png" /> be two Banach spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302203.png" /> denote the Banach space of all continuous (bounded) operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302204.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302205.png" /> (cf. also [[Banach space|Banach space]]; [[Continuous operator|Continuous operator]]). For an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302206.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302207.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302208.png" /> be the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s1302209.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022010.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022011.png" /> be the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022013.png" /> denotes the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022014.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022015.png" /> is a semi-Fredholm operator if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022016.png" /> is closed (i.e. it is a [[Normally-solvable operator|normally-solvable operator]]) and at least one of the vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022018.png" /> is of finite dimension. (The definition is partially redundant, since if the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022019.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022020.png" /> is closed.)

For a semi-Fredholm operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022021.png" />, its index, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022022.png" /></td> </tr></table>

is uniquely determined either as an integer, or as plus or minus infinity. In the first case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022023.png" /> is a [[Fredholm-operator(2)|Fredholm operator]]. Cf. also [[Index of an operator|Index of an operator]].

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022024.png" /> of all semi-Fredholm operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022025.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022026.png" /> and the index is constant on each connected component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022027.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022028.png" /> is a [[Compact operator|compact operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022030.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022032.png" /> is also in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022033.png" /> and its index equals that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022034.png" />. 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 [[#References|[a1]]] and T. Kato [[#References|[a2]]], who also treated the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130220/s13022035.png" /> is unbounded.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Kato,   "Perturbation theory for nullity, deficiency and other quantities of linear operators"  ''J. d'Anal. Math.'' , '''6'''  (1958)  pp. 261–322</TD></TR></table>