Editing Fredholm operator (section)

==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.