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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
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() 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
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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
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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) |
Wiener-Hopf operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-Hopf_operator&oldid=23145