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Difference between revisions of "Index of an operator"

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The difference between the dimensions of the deficiency subspaces (cf. [[Deficiency subspace|Deficiency subspace]]) of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050660/i0506601.png" />, that is, between those of its kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050660/i0506602.png" /> and its cokernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050660/i0506603.png" />, if these spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050660/i0506604.png" />.
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The difference between the dimensions of the deficiency subspaces (cf. [[Deficiency subspace|Deficiency subspace]]) of a linear operator $A\colon L_0\to L_1$, that is, between those of its kernel $\operatorname{Ker}A=A^{-1}(0)$ and its cokernel $\operatorname{Coker}A=L_1/A(L_0)$, if these spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation $Ax=b$.
  
 
====Comments====
 
====Comments====
The index defined above is also called the analytic index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050660/i0506605.png" />, cf. [[Index formulas|Index formulas]].
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The index defined above is also called the analytic index of $A$, cf. [[Index formulas|Index formulas]].
  
 
An important case, in which the index is well defined and is a homotopy invariant, is that of elliptic partial differential operators acting on sections of vector bundles over compact manifolds.
 
An important case, in which the index is well defined and is a homotopy invariant, is that of elliptic partial differential operators acting on sections of vector bundles over compact manifolds.

Latest revision as of 18:19, 21 November 2018

The difference between the dimensions of the deficiency subspaces (cf. Deficiency subspace) of a linear operator $A\colon L_0\to L_1$, that is, between those of its kernel $\operatorname{Ker}A=A^{-1}(0)$ and its cokernel $\operatorname{Coker}A=L_1/A(L_0)$, if these spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation $Ax=b$.

Comments

The index defined above is also called the analytic index of $A$, cf. Index formulas.

An important case, in which the index is well defined and is a homotopy invariant, is that of elliptic partial differential operators acting on sections of vector bundles over compact manifolds.

One can also define the index of, e.g., a linear Fredholm operator between Banach spaces, of an elliptic boundary value problem and of an "almost" pseudo-differential operator (cf. also [a1]).

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)
How to Cite This Entry:
Index of an operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Index_of_an_operator&oldid=12681