# 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 $\text{l} ^ { 2 }$ with matrix of the form $( \gamma _ { j - k } ) _ { j , k \geq 0 }$. The following boundedness criterion was obtained by P.R. Halmos (see [a1], [a5]): Let $\{ \gamma _ { j } \} _ { j \in \mathbf Z }$ be a sequence of complex numbers and let $T$ be the operator on $\text{l} ^ { 2 }$ with matrix $( \gamma _ { j - k } ) _ { j , k \geq 0 }$. Then $T$ is bounded if and only if there exists a function $\phi \in L ^ { \infty }$ on the unit circle $\bf T$ such that

\begin{equation*} \gamma _ { j } = \widehat { \phi } ( j ) , j \in \mathbf{Z}, \end{equation*}

where the $\hat { \phi } ( j )$, $j \in \mathbf{Z}$, are the Fourier coefficients of $\phi$ (cf. also Fourier series).

This theorem allows one to consider the following realization of Toeplitz operators on the Hardy class $H ^ { 2 }$ (cf. also Hardy classes). Let $\phi \in L ^ { \infty }$. One defines the Toeplitz operator $T _ { \phi } : H ^ { 2 } \rightarrow H ^ { 2 }$ by $T_{\phi}\,f = \mathcal{P}_{ +} \phi f$, where $\mathcal{P} _ { + }$ is the orthogonal projection onto $H ^ { 2 }$. The function $\phi$ is called the symbol of $T _ { \phi }$.

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 $k \in L ^ { 1 } ( \mathbf{R} )$ one can define the Wiener–Hopf operator $W _ { k }$ on $L ^ { 2 } ( \mathbf{R} _ { + } )$ by

\begin{equation*} ( W _ { k } f ) ( t ) = \int _ { 0 } ^ { \infty } k ( t - s ) f ( s ) d s , t \in {\bf R} _ { + }. \end{equation*}

Then $\| W _ { k } \| = \| \mathcal{F} k \| _ { L^\infty } $, where $\mathcal{F}$ is the Fourier transform. The definition of Wiener–Hopf operators can be extended to the case when $k$ is a tempered distribution whose Fourier transform is in $L^{\infty}$. In this case, $W _ { k }$ is unitarily equivalent to the Toeplitz operator $T _ { \phi }$, where $\phi = ( \mathcal{F} k ) \circ \text{o}$ and $\text{o}$ is a conformal mapping from the unit disc onto the upper half-plane.

The mapping $\phi \mapsto T _ { \phi }$ defined on $L^{\infty}$ is linear but not multiplicative. In fact, $T _ { \phi \psi } = T _ { \phi } T _ { \psi }$ if and only if $\psi \in H ^ { \infty }$ or $\overline { \phi } \in H ^ { \infty }$ (Halmos' theorem, see [a1]). It is easy to see that $T _ { \phi } ^ { * } = T _ { \overline { \phi } }$.

It is important in applications to be able to solve Toeplitz equations $T _ { \phi } f = g$. Therefore one of the most important problems in the study of Toeplitz operators is to describe the spectrum $\sigma ( T _ { \phi } )$ and the essential spectrum $\sigma _ { e } ( T _ { \phi } )$ (cf. also Spectrum of an operator).

Unlike the case of arbitrary operators, a Toeplitz operator $T _ { \phi }$ is invertible if and only if it is Fredholm and its index $\operatorname{ind} T _ { \phi } = \operatorname { dim } \operatorname { Ker } T _ { \phi } - \operatorname { dim } \operatorname { Ker } T _ { \phi } ^ { * } = 0$. This is a consequence of the following lemma, which is due to L.A.. Coburn ([a1]): If $\phi$ is a non-zero function in $L^{\infty}$, then either $\operatorname{Ker} T _ { \phi } = \{ 0 \}$ or $\operatorname { Ker } T _ { \phi } ^ { * } = \{ 0 \}$.

Hence,

\begin{equation*} \sigma ( T _ { \phi } ) = \sigma _ { \operatorname{e} } ( T _ { \phi } ) \bigcup \{ \lambda \notin \sigma _ { \operatorname{e} } ( T _ { \phi } ) : \text { ind } T _ { \phi - \lambda } \neq 0 \}. \end{equation*}

The following elementary results can be found in [a1].

If $\phi \in H ^ { \infty }$, then $\sigma ( T _ { \phi } )$ is the closure of $\phi ( D )$, where $D$ is the open unit disc (Wintner's theorem). If $\phi \in L ^ { \infty }$, then

\begin{equation} \tag{a1} \mathcal{R} ( \phi ) \subset \sigma _ { e } ( T _ { \phi } ) \subset \sigma ( T _ { \phi } ) \subset \operatorname { conv } ( \mathcal{R} ( \phi ) ). \end{equation}

Here, $\mathcal{R} ( \phi )$ is the essential range of $\phi$ and $\operatorname{conv} ( E )$ is the convex hull of a set $E$. 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 $\phi$ is a real function in $L^{\infty}$, then

\begin{equation*} \sigma ( T _ { \phi } ) = \operatorname { conv } ( \mathcal{R} ( \phi ) ) = [ \operatorname { essinf } \phi , \operatorname { esssup } \phi ]. \end{equation*}

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 $1$ almost everywhere on $\bf T$. Namely, $T _ { \phi }$ is invertible if and only if $\phi$ is invertible in $L^{\infty}$ and the operator $ T _ { \phi / | \phi | }$ is invertible.

The following theorem is due to A. Devinatz, H. Widom and N.K. Nikol'skii, see [a1], [a5]: Let $u$ be a unimodular function on $\bf T$. Then

i) $T _ { u }$ is left invertible if and only if $\operatorname { dist } _ { L^\infty } ( u , H ^ { \infty } ) < 1$;

ii) $T _ { u }$ is right invertible if and only if $\operatorname { dist } _ { L ^ \infty } ( \overline { u } , H ^ { \infty } ) < 1$;

iii) if $T _ { u }$ is invertible and there exists a function $h \in H ^ { \infty }$ such that $\| u - h \| _ { L } \infty < 1$, then $h$ is invertible in $H ^ { \infty }$;

iv) $T _ { u }$ is invertible if and only if there exists an outer function (cf. also Hardy classes) $h \in H ^ { \infty }$ such that $\| u - h \| _ { L } \infty < 1$;

v) if $T _ { u }$ is left invertible, then $T _ { u }$ is invertible if and only if $T _ { z u}$ is not left invertible.

The following invertibility criterion was obtained independently by Widom and Devinatz, see [a1]: Let $\phi \in L ^ { \infty }$. Then $T _ { \phi }$ is invertible if and only if $\phi$ is invertible in $L^{\infty}$ and the unimodular function $\phi / | \phi |$ admits a representation

\begin{equation*} \frac { \phi } { | \phi | } = \operatorname { exp } ( \xi + \widetilde { \eta } + c ), \end{equation*}

where $\xi $ and $ \eta $ are real functions in $L^{\infty}$, $c \in \mathbf R$, and $\tilde { \eta }$ is the harmonic conjugate of $ \eta $ (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 $\sigma ( T _ { \phi } )$ and improved by R.G. Douglas for $\sigma _ { e } ( T _ { \phi } )$ (see [a1]): Let $\phi \in L ^ { \infty }$. Then $\sigma _ { e } ( T _ { \phi } )$ is a connected set. Consequently, $\sigma ( T _ { \phi } )$ is connected.

There is no geometric description of the spectrum of a general Toeplitz operator. However, for certain classes of functions $\phi$ there exist nice geometric descriptions (see [a1]). For instance, let $\phi \in C ( \mathbf{T} )$. Then $\sigma _ { e } ( T _ { \phi } ) = \phi ( \mathbf{T} )$. If $\lambda \notin \phi ( \mathbf{T} )$, then

\begin{equation*} \operatorname{ind} T _ { \phi - \lambda } = - \text { wind } ( \phi - \lambda ) \end{equation*}

where $\operatorname{wind}\, f$ is the winding number of $f$ with respect to the origin.

A similar result holds if $\phi$ belongs to the algebra $H ^ { \infty } + C = \{ f + g : f \in C ( \mathbf{T} ) , g \in H ^ { \infty } \}$ (Douglas' theorem, see [a1]): Let $\phi \in H ^ { \infty } + C$; then $T _ { \phi }$ is a Fredholm operator if and only if $\phi$ is invertible in $H ^ { \infty } + C$. If $T _ { \phi }$ is Fredholm, then

\begin{equation*} \operatorname{ind}T _ { \phi } = -\operatorname{wind} \phi. \end{equation*}

Note that if $\phi$ is invertible in $H ^ { \infty } + C$, then its harmonic extension to the unit disc $D$ is separated away from $0$ near the boundary $\bf T$ and $\operatorname{wind} \phi$ is, by definition, the winding number of the restriction of the harmonic extension of $\phi$ to a circle of radius sufficiently close to $1$.

There is a similar geometric description of $\sigma ( T _ { \phi } )$ for piecewise-continuous functions $\phi$ (the Devinatz–Widom theorem, see [a1]). In this case, instead of considering the curve $\phi$ one has to consider the curve obtained from $\phi$ by adding intervals that join the points $\operatorname { lim } _ { t \rightarrow 0 ^ { + } } \phi ( e ^ { i t } \zeta )$ and $\operatorname { lim } _ { t \rightarrow 0^{-} } \phi ( e ^ { i t } \zeta )$.

There are several local principles in the theory of Toeplitz operators. For $\phi , \psi \in L ^ { \infty }$, the local distance at $\lambda \in \bf{T}$ is defined by

\begin{equation*} \operatorname { dist } _ { \lambda } ( \phi , \psi ) = \operatorname { limsup } _ { \zeta \rightarrow \lambda } | \phi ( \zeta ) - \psi ( \zeta ) |. \end{equation*}

The Simonenko local principle (see [a5]) is as follows. Let $\phi \in L ^ { \infty }$. Suppose that for each $\lambda \in \bf{T}$ there exists a $\phi _ { \lambda } \in L ^ { \infty }$ such that $T _ { \phi _ { \lambda } }$ is Fredholm and $\operatorname { dist } _ { \lambda } ( \phi , \phi _ { \lambda } ) = 0$. Then $T _ { \phi }$ is Fredholm.

See [a1] for the Douglas localization principle.

If $\phi$ is a real $L^{\infty}$-function, the self-adjoint Toeplitz operator has absolutely continuous spectral measure ([a6]). In [a3] and [a7] an explicit description of the spectral type of $T _ { \phi }$ is given for $\phi \in L ^ { \infty }$.

It is important in applications to study vectorial Toeplitz operators $T _ { \Phi }$ with matrix-valued symbols $\Phi$. There are vectorial Fredholm Toeplitz operators $T _ { \Phi }$ with zero index which are not invertible. If $\Phi$ is a continuous matrix-valued function, then $T _ { \Phi }$ is Fredholm if and only if $\operatorname{det} \Phi$ is invertible in $C ( \mathbf{T} )$ and

\begin{equation*} \operatorname { ind }T_{\Phi} = -\operatorname {wind} \operatorname {det} \Phi . \end{equation*}

Similar results are valid for matrix-valued functions in $H ^ { \infty } + C$ 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 $\Phi$ be an $( n \times n )$-matrix-valued $L^{\infty}$ function on $\bf T$. Then $T _ { \Phi }$ is Fredholm if and only if $\Phi$ admits a factorization

\begin{equation*} \Phi = \Psi _ { 2 } ^ { * } \wedge \Psi _ { 1 }, \end{equation*}

where $\Psi _ { 1 }$ and $\Psi _ { 2 }$ are matrix functions invertible in $H ^ { 2 }$,

\begin{equation*} \Lambda = \left( \begin{array} { c c c c } { z ^ { k _ { 1 } } } & { 0 } & { \ldots } & { 0 } \\ { 0 } & { z ^ { k_{2} } } & { \ldots } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { 0 } & { \ldots } & { z ^ { k _ { n } } } \end{array} \right) , k _ { 1 } , \ldots , k _ { n } \in \mathbf{Z}, \end{equation*}

and the operator $B$, defined on the set of polynomials in $H ^ { 2 } ( \mathbf{C} ^ { n } )$ by

\begin{equation*} B f = \Psi _ { 2 } ^ { - 1 } \mathcal{P} _ { + } \overline { \Lambda } \mathcal{P} _ { + } \overline { \Psi }^ {-1}_{1} f, \end{equation*}

extends to a bounded operator on $H ^ { 2 } ( \mathbf{C} ^ { n } )$.

#### 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 |

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Toeplitz operator.

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