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An operator which arises from another operator by compression to a subspace. Given a [[Linear operator|linear operator]]  $  A : X \rightarrow X $
and a projection  $  P = P  ^ {2} $
on  $  X $
with range  $  { \mathop{\rm Im} } P $, 
the corresponding Wiener–Hopf operator  $  W _ {P} ( A ) $
is defined as the operator on  $  { \mathop{\rm Im} } P $
that sends  $  x \in { \mathop{\rm Im} } P $
to  $  P ( Ax ) \in { \mathop{\rm Im} } P $. 
Thus,  $  W _ {P} ( A ) = PA \mid  _ { { \mathop{\rm Im}  } P } $.

A Wiener–Hopf integral operator formally acts by the rule

$$ 
( Wf ) ( x ) = cf ( x ) + \int\limits _ { 0 } ^  \infty   {k ( x - t ) f ( t ) }  {dt }
$$

( $  x > 0 $) 
on some space of functions over  $  \mathbf R _ {+} = ( 0, \infty ) $, 
say on  $  L _ {p} ( \mathbf R _ {+} ) $(
$  1 \leq  p \leq  \infty $). 
It may be regarded as the restriction to  $  L _ {p} ( \mathbf R _ {+} ) $
of a convolution integral operator on  $  L _ {p} ( \mathbf R ) $. 
The operator  $  W $
is bounded if, for example,  $  c \in \mathbf C $
and  $  k \in L _ {1} ( \mathbf R ) $. 
Many properties of  $  W $
can be read off from its symbol. This is the function  $  a $
given by

$$ 
a ( \xi ) = c + \int\limits _ { \mathbf R } {k ( t ) e ^ {i \xi t } }  {dt } \quad ( \xi \in \mathbf R ) .
$$

The operator  $  W $
is Fredholm (cf. [[Fredholm-operator(2)|Fredholm operator]]), i.e. invertible modulo compact operators, if and only if its symbol  $  a $
has no zeros on the one-point compactification  $  \mathbf R \cup \{ \infty \} $(
cf. also [[Aleksandrov compactification|Aleksandrov compactification]]) of  $  \mathbf R $. 
In that case the kernel and cokernel dimensions of  $  W $
are:

$$ 
{ \mathop{\rm dim} } { \mathop{\rm Ker} } W =  \max  \{ - \kappa, 0 \}
$$

$$ 
{ \mathop{\rm dim} } { \mathop{\rm Coker} } W =  \max  \{ \kappa, 0 \} ,
$$

where  $  \kappa $
is the [[Winding number|winding number]] of the symbol  $  a $
about the origin. The equation  $  Wf = g $
can be solved by Wiener–Hopf factorization, which means that one represents  $  a $
in the form

$$ 
a ( \xi ) = a _ {-} ( \xi ) \left ( {
\frac{\xi - i }{\xi + i }
 } \right )  ^  \kappa  a _ {+} ( \xi )
$$

such that  $  a _ {-} $
and  $  a _ {+} $
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  $  S _ {+} $(
cf. also [[Singular integral|Singular integral]]) on  $  L _ {p} ( \mathbf R _ {+} ) $(
$  1 < p < \infty $),

$$ 
( S _ {+} f ) ( x ) = {
\frac{1}{\pi i }
 } \int\limits _ {\mathbf R _ {+} } { {
\frac{f ( t ) }{t - x }
 } }  {dt } \quad ( x \in \mathbf R _ {+} ) ,
$$

can be interpreted as the Wiener–Hopf integral operator with symbol  $  - { \mathop{\rm sign} } \xi $. 
The spectrum of  $  S _ {+} $
on  $  L _ {p} ( \mathbf R _ {+} ) $
is the set of all  $  \lambda \in \mathbf C $
at which the line segment  $  [ - 1,1 ] $
is seen at an angle of at least  $   \max  \{ { {2 \pi } / p } , { {2 \pi } / q } \} $, 
where  $  {1 / p } + {1 / q } = 1 $.

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  $  ( a _ {j - k }  ) _ {j,k = 0 }   ^  \infty  $
on some sequence space, e.g. on  $  l  ^ {p} ( \mathbf Z _ {+} ) $. 
In this case the symbol is the function on the complex unit circle whose Fourier coefficients constitute the sequence  $  ( a _ {n} ) _ {n \in \mathbf Z }  $.

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>