View source for Fredholm spectrum

A [[Bounded operator|bounded operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201601.png" /> on a complex [[Hilbert space|Hilbert space]] is said to be essentially left invertible (respectively, essentially right invertible) if there exists a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201602.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201603.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201604.png" />) has finite rank. An operator that is essentially left or right invertible is also called a semi-Fredholm operator, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201605.png" /> is a [[Fredholm-operator(2)|Fredholm operator]] if it is both left and right essentially invertible. F.V. Atkinson proved that an operator is Fredholm if and only if it has closed range and the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201607.png" /> have finite dimension. For a semi-Fredholm operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f1201608.png" /> one defines the index

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this is an integer or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016010.png" />. The set of semi-Fredholm operators is open, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016011.png" /> is continuous (thus locally constant). Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016013.png" /> is a [[Compact operator|compact operator]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016014.png" /> is finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016015.png" /> is Fredholm. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016016.png" /> be the set of complex scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016018.png" /> is not semi-Fredholm; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016019.png" /> is a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016020.png" /> (cf. [[Spectrum of an operator|Spectrum of an operator]]). The difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016021.png" /> is called the semi-Fredholm spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016022.png" />, and a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016023.png" /> in this set belongs to the Fredholm spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016025.png" /> is Fredholm. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016026.png" /> is one of the connected components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016027.png" />, then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016028.png" /> is constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016029.png" />; call it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016032.png" /> is entirely contained in the semi-Fredholm spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016033.png" /> (in the Fredholm spectrum if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016034.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016035.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016036.png" /> is contained in the Fredholm spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016037.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016038.png" /> has no accumulation points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016039.png" />. More generally, I.C. Gohberg showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016040.png" /> is constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016041.png" />, with the possible exception of a countable set with no accumulation points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016042.png" />; the dimension is greater if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016043.png" /> is one of the exceptional points.

An interesting calculation of the Fredholm spectrum was done by Gohberg when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016044.png" /> is a Toeplitz operator (cf. also [[Toeplitz matrix|Toeplitz matrix]]; [[Calderón–Toeplitz operator|Calderón–Toeplitz operator]]) whose symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016045.png" /> is a continuous function on the unit circle. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016046.png" /> is the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016048.png" /> equals minus the [[Winding number|winding number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016049.png" /> about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016050.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016051.png" />.

The presence of a semi-Fredholm spectrum is related with the notion of quasi-triangularity introduced by P.R. Halmos. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016052.png" /> is quasi-triangular if it can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016053.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016054.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016055.png" /> is triangular in some basis. R.G. Douglas and C.M. Pearcy showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016056.png" /> cannot be quasi-triangular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016057.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016058.png" /> in the semi-Fredholm spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016059.png" />. Quite surprisingly, the converse of this statement is also true, as shown by C. Apostol, C. Foiaş and D. Voiculescu. Subsequent developments involving the Fredholm spectrum include the approximation theory of Hilbert-space operators developed by Apostol, D. Herrero, and Voiculescu.

The notion of Fredholm spectrum can be extended to operators on other topological vector spaces, to unbounded operators, and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016060.png" />-tuples of commuting operators. A different generalization is to define the Fredholm spectrum for elements in von Neumann algebras. The algebra of bounded operators on a Hilbert space is a factor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016062.png" />. An appropriate notion of finite-rank element exists in any factor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120160/f12016063.png" />, and this leads to a corresponding notion of Fredholm operator and Fredholm spectrum. Analogues have been found in other Banach algebras as well.

The Fredholm property was also defined in a non-linear context by S. Smale. A differentiable mapping (cf. also [[Differentiation of a mapping|Differentiation of a mapping]]) between two open sets in a [[Banach space|Banach space]] is Fredholm if its derivative at every point is a linear [[Fredholm-operator(2)|Fredholm operator]]. This leads to the notion of a Fredholm mapping on an infinite-dimensional manifold. Smale used this notion to extend to infinite dimensions certain results of A. Sard and R. Thom.