View source for Wiener-Hopf operator

An operator which arises from another operator by compression to a subspace. Given a [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100501.png" /> and a projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100502.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100503.png" /> with range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100504.png" />, the corresponding Wiener–Hopf operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100505.png" /> is defined as the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100506.png" /> that sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100507.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100508.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w1100509.png" />.

A Wiener–Hopf integral operator formally acts by the rule

<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/w/w110/w110050/w11005010.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005011.png" />) on some space of functions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005012.png" />, say on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005014.png" />). It may be regarded as the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005015.png" /> of a convolution integral operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005016.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005017.png" /> is bounded if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005019.png" />. Many properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005020.png" /> can be read off from its symbol. This is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005021.png" /> given by

<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/w/w110/w110050/w11005022.png" /></td> </tr></table>

The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005023.png" /> is Fredholm (cf. [[Fredholm-operator(2)|Fredholm operator]]), i.e. invertible modulo compact operators, if and only if its symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005024.png" /> has no zeros on the one-point compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005025.png" /> (cf. also [[Aleksandrov compactification|Aleksandrov compactification]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005026.png" />. In that case the kernel and cokernel dimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005027.png" /> are:

<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/w/w110/w110050/w11005028.png" /></td> </tr></table>

<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/w/w110/w110050/w11005029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005030.png" /> is the [[Winding number|winding number]] of the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005031.png" /> about the origin. The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005032.png" /> can be solved by Wiener–Hopf factorization, which means that one represents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005033.png" /> in the form

<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/w/w110/w110050/w11005034.png" /></td> </tr></table>

such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005036.png" /> extend to analytic functions without zeros in the lower and upper complex half-planes, respectively.

Many interesting operators are Wiener–Hopf integral operators with discontinuous symbols. For example, the Cauchy singular integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005037.png" /> (cf. also [[Singular integral|Singular integral]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005038.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005039.png" />),

<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/w/w110/w110050/w11005040.png" /></td> </tr></table>

can be interpreted as the Wiener–Hopf integral operator with symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005041.png" />. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005043.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005044.png" /> at which the line segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005045.png" /> is seen at an angle of at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005047.png" />.

Wiener–Hopf integral operators with matrix-valued symbols, on finite intervals, or on higher-dimensional domains (including the quarter-plane) have also been extensively studied.

A discrete Wiener–Hopf operator, or a Toeplitz operator (cf. also [[Toeplitz matrix|Toeplitz matrix]]), is given by a matrix of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005048.png" /> on some sequence space, e.g. on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005049.png" />. In this case the symbol is the function on the complex unit circle whose Fourier coefficients constitute the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110050/w11005050.png" />.

There is a rich literature on Wiener–Hopf operators. A good introduction is the classical monograph [[#References|[a1]]]; [[#References|[a2]]] and [[#References|[a3]]] provide an overview of some recent developments.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Gohberg,   I.A. Feldman,   "Convolution equations and projection methods for their solution" , Amer. Math. Soc.  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Böttcher,   B. Silbermann,   "Analysis of Toeplitz operators" , Springer  (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Gohberg,   S. Goldberg,   M.A. Kaashoek,   "Classes of linear operators" , '''I–II''' , Birkhäuser  (1990–1993)</TD></TR></table>