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

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

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,

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

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

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

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

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

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

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

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