# Wiener-Hopf operator

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.

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. BÃ¶ttcher (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article