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.

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)