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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" />.
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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
 
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>
+
$$
 +
( Wf ) ( x ) = cf ( x ) + \int\limits _ { 0 } ^  \infty  {k ( x - t ) f ( t ) }  {dt }
 +
$$
  
(<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
+
( $  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
  
<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>
+
$$
 +
a ( \xi ) = c + \int\limits _ { \mathbf R } {k ( t ) e ^ {i \xi t } }  {dt } \quad ( \xi \in \mathbf R ) .
 +
$$
  
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:
+
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:
  
<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>
+
$$
 +
{ \mathop{\rm dim} } { \mathop{\rm Ker} } W = \max  \{ - \kappa, 0 \}
 +
$$
  
<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>
+
$$
 +
{ \mathop{\rm dim} } { \mathop{\rm Coker} } W = \max  \{ \kappa, 0 \} ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
a ( \xi ) = a _ {-} ( \xi ) \left ( {
 +
\frac{\xi - i }{\xi + i }
 +
} \right )  ^  \kappa  a _ {+} ( \xi )
 +
$$
  
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.
+
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 <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" />),
+
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 $),
  
<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>
+
$$
 +
( S _ {+} f ) ( x ) = {
 +
\frac{1}{\pi i }
 +
} \int\limits _ {\mathbf R _ {+} } { {
 +
\frac{f ( t ) }{t - x }
 +
} }  {dt } \quad ( x \in \mathbf R _ {+} ) ,
 +
$$
  
can be interpreted as the Wiener–Hopf integral operator with symbol <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" />.
+
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.
 
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" />.
+
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.
 
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.

Latest revision as of 08:29, 6 June 2020


An operator which arises from another operator by compression to a subspace. Given a 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), 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) 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 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) 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), 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 [a1]; [a2] and [a3] provide an overview of some recent developments.

References

[a1] I. Gohberg, I.A. Feldman, "Convolution equations and projection methods for their solution" , Amer. Math. Soc. (1974)
[a2] A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990)
[a3] I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators" , I–II , Birkhäuser (1990–1993)
How to Cite This Entry:
Wiener-Hopf operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-Hopf_operator&oldid=49217
This article was adapted from an original article by A. Böttcher (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article