Wiener-Hopf operator

An operator which arises from another operator by compression to a subspace. Given a linear operator and a projection on with range , the corresponding Wiener–Hopf operator is defined as the operator on that sends to . Thus, .

A Wiener–Hopf integral operator formally acts by the rule

() on some space of functions over , say on (). It may be regarded as the restriction to of a convolution integral operator on . The operator is bounded if, for example, and . Many properties of can be read off from its symbol. This is the function given by

The operator is Fredholm (cf. Fredholm operator), i.e. invertible modulo compact operators, if and only if its symbol has no zeros on the one-point compactification (cf. also Aleksandrov compactification) of . In that case the kernel and cokernel dimensions of are:

where is the winding number of the symbol about the origin. The equation can be solved by Wiener–Hopf factorization, which means that one represents in the form

such that and 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 (cf. also Singular integral) on (),

can be interpreted as the Wiener–Hopf integral operator with symbol . The spectrum of on is the set of all at which the line segment is seen at an angle of at least , where .

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 on some sequence space, e.g. on . In this case the symbol is the function on the complex unit circle whose Fourier coefficients constitute the sequence .

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)