Toeplitz operator
Together with the class of Hankel operators (cf. also Hankel operator), the class of Toeplitz operators is one of the most important classes of operators on Hardy spaces. A Toeplitz operator can be defined as an operator on with matrix of the form
. The following boundedness criterion was obtained by P.R. Halmos (see [a1], [a5]): Let
be a sequence of complex numbers and let
be the operator on
with matrix
. Then
is bounded if and only if there exists a function
on the unit circle
such that
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where the ,
, are the Fourier coefficients of
(cf. also Fourier series).
This theorem allows one to consider the following realization of Toeplitz operators on the Hardy class (cf. also Hardy classes). Let
. One defines the Toeplitz operator
by
, where
is the orthogonal projection onto
. The function
is called the symbol of
.
Toeplitz operators are important in many applications (prediction theory, boundary-value problems for analytic functions, singular integral equations). Toeplitz operators are unitarily equivalent to Wiener–Hopf operators (cf. also Wiener–Hopf operator). For a function one can define the Wiener–Hopf operator
on
by
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Then , where
is the Fourier transform. The definition of Wiener–Hopf operators can be extended to the case when
is a tempered distribution whose Fourier transform is in
. In this case,
is unitarily equivalent to the Toeplitz operator
, where
and
is a conformal mapping from the unit disc onto the upper half-plane.
The mapping defined on
is linear but not multiplicative. In fact,
if and only if
or
(Halmos' theorem, see [a1]). It is easy to see that
.
It is important in applications to be able to solve Toeplitz equations . Therefore one of the most important problems in the study of Toeplitz operators is to describe the spectrum
and the essential spectrum
(cf. also Spectrum of an operator).
Unlike the case of arbitrary operators, a Toeplitz operator is invertible if and only if it is Fredholm and its index
. This is a consequence of the following lemma, which is due to L.A.. Coburn ([a1]): If
is a non-zero function in
, then either
or
.
Hence,
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The following elementary results can be found in [a1].
If , then
is the closure of
, where
is the open unit disc (Wintner's theorem). If
, then
![]() | (a1) |
Here, is the essential range of
and
is the convex hull of a set
. Note that (a1) is a combination of an improvement of a Hartman–Wintner theorem and a Brown–Halmos theorem.
The following theorem, which is also due to P. Hartman and A. Wintner, describes the spectrum of self-adjoint Toeplitz operators (see [a1]): If is a real function in
, then
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The problem of the invertibility of an arbitrary Toeplitz operator can be reduced to the case when the symbol is unimodular, i.e., has modulus almost everywhere on
. Namely,
is invertible if and only if
is invertible in
and the operator
is invertible.
The following theorem is due to A. Devinatz, H. Widom and N.K. Nikol'skii, see [a1], [a5]: Let be a unimodular function on
. Then
i) is left invertible if and only if
;
ii) is right invertible if and only if
;
iii) if is invertible and there exists a function
such that
, then
is invertible in
;
iv) is invertible if and only if there exists an outer function (cf. also Hardy classes)
such that
;
v) if is left invertible, then
is invertible if and only if
is not left invertible.
The following invertibility criterion was obtained independently by Widom and Devinatz, see [a1]: Let . Then
is invertible if and only if
is invertible in
and the unimodular function
admits a representation
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where and
are real functions in
,
, and
is the harmonic conjugate of
(cf. also Conjugate function).
Note that this theorem is equivalent to the Helson–Szegö theorem on weighted boundedness of the harmonic conjugation operator.
The following general result was obtained by Widom for and improved by R.G. Douglas for
(see [a1]): Let
. Then
is a connected set. Consequently,
is connected.
There is no geometric description of the spectrum of a general Toeplitz operator. However, for certain classes of functions there exist nice geometric descriptions (see [a1]). For instance, let
. Then
. If
, then
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where is the winding number of
with respect to the origin.
A similar result holds if belongs to the algebra
(Douglas' theorem, see [a1]): Let
; then
is a Fredholm operator if and only if
is invertible in
. If
is Fredholm, then
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Note that if is invertible in
, then its harmonic extension to the unit disc
is separated away from
near the boundary
and
is, by definition, the winding number of the restriction of the harmonic extension of
to a circle of radius sufficiently close to
.
There is a similar geometric description of for piecewise-continuous functions
(the Devinatz–Widom theorem, see [a1]). In this case, instead of considering the curve
one has to consider the curve obtained from
by adding intervals that join the points
and
.
There are several local principles in the theory of Toeplitz operators. For , the local distance at
is defined by
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The Simonenko local principle (see [a5]) is as follows. Let . Suppose that for each
there exists a
such that
is Fredholm and
. Then
is Fredholm.
See [a1] for the Douglas localization principle.
If is a real
-function, the self-adjoint Toeplitz operator has absolutely continuous spectral measure ([a6]). In [a3] and [a7] an explicit description of the spectral type of
is given for
.
It is important in applications to study vectorial Toeplitz operators with matrix-valued symbols
. There are vectorial Fredholm Toeplitz operators
with zero index which are not invertible. If
is a continuous matrix-valued function, then
is Fredholm if and only if
is invertible in
and
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Similar results are valid for matrix-valued functions in and for piecewise-continuous matrix-valued functions (see [a2]).
The following Simonenko theorem (see [a4]) gives a criterion for vectorial Toeplitz operators to be Fredholm. Let be an
-matrix-valued
function on
. Then
is Fredholm if and only if
admits a factorization
![]() |
where and
are matrix functions invertible in
,
![]() |
and the operator , defined on the set of polynomials in
by
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extends to a bounded operator on .
References
[a1] | R.G. Douglas, "Banach algebra techniques in operator theory" , Acad. Press (1972) |
[a2] | R.G. Douglas, "Banach algebra techniques in the theory of Toeplitz operators" , CBMS , 15 , Amer. Math. Soc. (1973) |
[a3] | R.S. Ismagilov, "On the spectrum of Toeplitz matrices" Dokl. Akad. Nauk SSSR , 149 (1963) pp. 769–772 |
[a4] | G.S. Litvinchuk, I.M. Spitkovski, "Factorization of measurable matrix functions" , Oper. Th. Adv. Appl. , 25 , Birkhäuser (1987) |
[a5] | N.K. Nikol'skii, "Treatise on the shift operator" , Springer (1986) |
[a6] | M. Rosenblum, "The absolute continuity of Toeplitz's matrices" Pacific J. Math. , 10 (1960) pp. 987–996 |
[a7] | M. Rosenblum, "A concrete spectral theory for self-adjoint Toeplitz operators" Amer. J. Math. , 87 (1965) pp. 709–718 |
Toeplitz operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_operator&oldid=50002