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Together with the class of Hankel operators (cf. also [[Hankel operator|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201401.png" /> with [[Matrix|matrix]] of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201402.png" />. The following boundedness criterion was obtained by P.R. Halmos (see [[#References|[a1]]], [[#References|[a5]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201403.png" /> be a sequence of complex numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201404.png" /> be the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201405.png" /> with matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201406.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201407.png" /> is bounded if and only if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201408.png" /> on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t1201409.png" /> such that
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014010.png" /></td> </tr></table>
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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014012.png" />, are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014013.png" /> (cf. also [[Fourier series|Fourier series]]).
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This theorem allows one to consider the following realization of Toeplitz operators on the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014014.png" /> (cf. also [[Hardy classes|Hardy classes]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014015.png" />. One defines the Toeplitz operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014018.png" /> is the orthogonal projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014019.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014020.png" /> is called the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014021.png" />.
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Together with the class of Hankel operators (cf. also [[Hankel operator|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|matrix]] of the form $( \gamma _ { j - k } ) _ { j , k \geq 0 }$. The following boundedness criterion was obtained by P.R. Halmos (see [[#References|[a1]]], [[#References|[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
  
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|Wiener–Hopf operator]]). For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014022.png" /> one can define the Wiener–Hopf operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014024.png" /> by
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\begin{equation*} \gamma _ { j } = \widehat { \phi } ( j ) , j \in \mathbf{Z}, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014025.png" /></td> </tr></table>
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where the $\hat { \phi } ( j )$, $j \in \mathbf{Z}$, are the Fourier coefficients of $\phi$ (cf. also [[Fourier series|Fourier series]]).
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014027.png" /> is the [[Fourier transform|Fourier transform]]. The definition of Wiener–Hopf operators can be extended to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014028.png" /> is a tempered distribution whose Fourier transform is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014029.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014030.png" /> is unitarily equivalent to the Toeplitz operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014033.png" /> is a [[Conformal mapping|conformal mapping]] from the unit disc onto the upper half-plane.
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This theorem allows one to consider the following realization of Toeplitz operators on the Hardy class $H ^ { 2 }$ (cf. also [[Hardy classes|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 $T _ { \phi }$.
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014034.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014035.png" /> is linear but not multiplicative. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014036.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014037.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014038.png" /> (Halmos' theorem, see [[#References|[a1]]]). It is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014039.png" />.
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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|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
  
It is important in applications to be able to solve Toeplitz equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014040.png" />. Therefore one of the most important problems in the study of Toeplitz operators is to describe the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014041.png" /> and the essential spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014042.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]).
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\begin{equation*} ( W _ { k } f ) ( t ) = \int _ { 0 } ^ { \infty } k ( t - s ) f ( s ) d s , t \in {\bf R} _ { + }. \end{equation*}
  
Unlike the case of arbitrary operators, a Toeplitz operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014043.png" /> is invertible if and only if it is Fredholm and its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014044.png" />. This is a consequence of the following lemma, which is due to L.A.. Coburn ([[#References|[a1]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014045.png" /> is a non-zero function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014046.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014047.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014048.png" />.
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Then $\| W _ { k } \| = \| \mathcal{F} k \| _ { L^\infty } $, where $\mathcal{F}$ is the [[Fourier transform|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|conformal mapping]] from the unit disc onto the upper half-plane.
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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 [[#References|[a1]]]). It is easy to see that $T _ { \phi } ^ { * } = T _ { \overline { \phi } }$.
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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|Spectrum of an operator]]).
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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 ([[#References|[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,
 
Hence,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014049.png" /></td> </tr></table>
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\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 [[#References|[a1]]].
 
The following elementary results can be found in [[#References|[a1]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014051.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014053.png" /> is the open unit disc (Wintner's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014054.png" />, then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014056.png" /> is the essential range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014058.png" /> is the convex hull of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014059.png" />. Note that (a1) is a combination of an improvement of a Hartman–Wintner theorem and a Brown–Halmos theorem.
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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 [[#References|[a1]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014060.png" /> is a real function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014061.png" />, then
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The following theorem, which is also due to P. Hartman and A. Wintner, describes the spectrum of self-adjoint Toeplitz operators (see [[#References|[a1]]]): If $\phi$ is a real function in $L^{\infty}$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014062.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014063.png" /> almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014064.png" />. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014065.png" /> is invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014066.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014067.png" /> and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014068.png" /> is invertible.
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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 $1$ 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 [[#References|[a1]]], [[#References|[a5]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014069.png" /> be a unimodular function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014070.png" />. Then
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The following theorem is due to A. Devinatz, H. Widom and N.K. Nikol'skii, see [[#References|[a1]]], [[#References|[a5]]]: Let $u$ be a unimodular function on $\bf T$. Then
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014071.png" /> is left invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014072.png" />;
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i) $T _ { u }$ is left invertible if and only if $\operatorname { dist } _ { L^\infty }  ( u , H ^ { \infty } ) < 1$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014073.png" /> is right invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014074.png" />;
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ii) $T _ { u }$ is right invertible if and only if $\operatorname { dist } _ { L ^ \infty } ( \overline { u } , H ^ { \infty } ) < 1$;
  
iii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014075.png" /> is invertible and there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014076.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014077.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014078.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014079.png" />;
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014080.png" /> is invertible if and only if there exists an outer function (cf. also [[Hardy classes|Hardy classes]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014082.png" />;
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iv) $T _ { u }$ is invertible if and only if there exists an outer function (cf. also [[Hardy classes|Hardy classes]]) $h \in H ^ { \infty }$ such that $\| u - h \| _ { L } \infty < 1$;
  
v) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014083.png" /> is left invertible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014084.png" /> is invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014085.png" /> is not left invertible.
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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 [[#References|[a1]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014086.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014087.png" /> is invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014088.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014089.png" /> and the unimodular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014090.png" /> admits a representation
+
The following invertibility criterion was obtained independently by Widom and Devinatz, see [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014091.png" /></td> </tr></table>
+
\begin{equation*} \frac { \phi } { | \phi | } = \operatorname { exp } ( \xi + \widetilde { \eta } + c ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014093.png" /> are real functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014096.png" /> is the harmonic conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014097.png" /> (cf. also [[Conjugate function|Conjugate function]]).
+
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|Conjugate function]]).
  
 
Note that this theorem is equivalent to the Helson–Szegö theorem on weighted boundedness of the harmonic conjugation operator.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014098.png" /> and improved by R.G. Douglas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t12014099.png" /> (see [[#References|[a1]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140100.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140101.png" /> is a connected set. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140102.png" /> is connected.
+
The following general result was obtained by Widom for $\sigma ( T _ { \phi } )$ and improved by R.G. Douglas for $\sigma _ { e } ( T _ { \phi } )$ (see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140103.png" /> there exist nice geometric descriptions (see [[#References|[a1]]]). For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140104.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140105.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140106.png" />, then
+
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 [[#References|[a1]]]). For instance, let $\phi \in C ( \mathbf{T} )$. Then $\sigma _ { e } ( T _ { \phi } ) = \phi ( \mathbf{T} )$. If $\lambda \notin \phi ( \mathbf{T} )$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140107.png" /></td> </tr></table>
+
\begin{equation*} \operatorname{ind} T _ { \phi - \lambda } = - \text { wind } ( \phi - \lambda ) \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140108.png" /> is the [[Winding number|winding number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140109.png" /> with respect to the origin.
+
where $\operatorname{wind}\, f$ is the [[Winding number|winding number]] of $f$ with respect to the origin.
  
A similar result holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140110.png" /> belongs to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140111.png" /> (Douglas' theorem, see [[#References|[a1]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140112.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140113.png" /> is a [[Fredholm-operator(2)|Fredholm operator]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140114.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140115.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140116.png" /> is Fredholm, then
+
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 [[#References|[a1]]]): Let $\phi \in H ^ { \infty } + C$; then $T _ { \phi }$ is a [[Fredholm-operator(2)|Fredholm operator]] if and only if $\phi$ is invertible in $H ^ { \infty } + C$. If $T _ { \phi }$ is Fredholm, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140117.png" /></td> </tr></table>
+
\begin{equation*} \operatorname{ind}T _ { \phi } = -\operatorname{wind} \phi. \end{equation*}
  
Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140118.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140119.png" />, then its harmonic extension to the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140120.png" /> is separated away from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140121.png" /> near the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140123.png" /> is, by definition, the winding number of the restriction of the harmonic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140124.png" /> to a circle of radius sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140125.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140126.png" /> for piecewise-continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140127.png" /> (the Devinatz–Widom theorem, see [[#References|[a1]]]). In this case, instead of considering the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140128.png" /> one has to consider the curve obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140129.png" /> by adding intervals that join the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140131.png" />.
+
There is a similar geometric description of $\sigma ( T _ { \phi } )$ for piecewise-continuous functions $\phi$ (the Devinatz–Widom theorem, see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140132.png" />, the local distance at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140133.png" /> is defined by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140134.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { dist } _ { \lambda } ( \phi , \psi ) = \operatorname { limsup } _ { \zeta \rightarrow \lambda } | \phi ( \zeta ) - \psi ( \zeta ) |. \end{equation*}
  
The Simonenko local principle (see [[#References|[a5]]]) is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140135.png" />. Suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140136.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140137.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140138.png" /> is Fredholm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140139.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140140.png" /> is Fredholm.
+
The Simonenko local principle (see [[#References|[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 [[#References|[a1]]] for the Douglas localization principle.
 
See [[#References|[a1]]] for the Douglas localization principle.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140141.png" /> is a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140142.png" />-function, the self-adjoint Toeplitz operator has absolutely continuous spectral measure ([[#References|[a6]]]). In [[#References|[a3]]] and [[#References|[a7]]] an explicit description of the spectral type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140143.png" /> is given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140144.png" />.
+
If $\phi$ is a real $L^{\infty}$-function, the self-adjoint Toeplitz operator has absolutely continuous spectral measure ([[#References|[a6]]]). In [[#References|[a3]]] and [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140145.png" /> with matrix-valued symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140146.png" />. There are vectorial Fredholm Toeplitz operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140147.png" /> with zero index which are not invertible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140148.png" /> is a continuous matrix-valued function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140149.png" /> is Fredholm if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140150.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140151.png" /> and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140152.png" /></td> </tr></table>
+
\begin{equation*}  \operatorname { ind }T_{\Phi} = -\operatorname {wind} \operatorname {det} \Phi . \end{equation*}
  
Similar results are valid for matrix-valued functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140153.png" /> and for piecewise-continuous matrix-valued functions (see [[#References|[a2]]]).
+
Similar results are valid for matrix-valued functions in $H ^ { \infty } + C$ and for piecewise-continuous matrix-valued functions (see [[#References|[a2]]]).
  
The following Simonenko theorem (see [[#References|[a4]]]) gives a criterion for vectorial Toeplitz operators to be Fredholm. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140154.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140155.png" />-matrix-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140156.png" /> function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140157.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140158.png" /> is Fredholm if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140159.png" /> admits a factorization
+
The following Simonenko theorem (see [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140160.png" /></td> </tr></table>
+
\begin{equation*} \Phi = \Psi _ { 2 } ^ { * } \wedge \Psi _ { 1 }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140161.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140162.png" /> are matrix functions invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140163.png" />,
+
where $\Psi _ { 1 }$ and $\Psi _ { 2 }$ are matrix functions invertible in $H ^ { 2 }$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140164.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140165.png" />, defined on the set of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140166.png" /> by
+
and the operator $B$, defined on the set of polynomials in $H ^ { 2 } ( \mathbf{C} ^ { n } )$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140167.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120140/t120140168.png" />.
+
extends to a bounded operator on $H ^ { 2 } ( \mathbf{C} ^ { n } )$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.G. Douglas,  "Banach algebra techniques in operator theory" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Douglas,  "Banach algebra techniques in the theory of Toeplitz operators" , ''CBMS'' , '''15''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.S. Ismagilov,  "On the spectrum of Toeplitz matrices"  ''Dokl. Akad. Nauk SSSR'' , '''149'''  (1963)  pp. 769–772</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.S. Litvinchuk,  I.M. Spitkovski,  "Factorization of measurable matrix functions" , ''Oper. Th. Adv. Appl.'' , '''25''' , Birkhäuser  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N.K. Nikol'skii,  "Treatise on the shift operator" , Springer  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Rosenblum,  "The absolute continuity of Toeplitz's matrices"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 987–996</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Rosenblum,  "A concrete spectral theory for self-adjoint Toeplitz operators"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 709–718</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  R.G. Douglas,  "Banach algebra techniques in operator theory" , Acad. Press  (1972)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.G. Douglas,  "Banach algebra techniques in the theory of Toeplitz operators" , ''CBMS'' , '''15''' , Amer. Math. Soc.  (1973)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R.S. Ismagilov,  "On the spectrum of Toeplitz matrices"  ''Dokl. Akad. Nauk SSSR'' , '''149'''  (1963)  pp. 769–772</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G.S. Litvinchuk,  I.M. Spitkovski,  "Factorization of measurable matrix functions" , ''Oper. Th. Adv. Appl.'' , '''25''' , Birkhäuser  (1987)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  N.K. Nikol'skii,  "Treatise on the shift operator" , Springer  (1986)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M. Rosenblum,  "The absolute continuity of Toeplitz's matrices"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 987–996</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. Rosenblum,  "A concrete spectral theory for self-adjoint Toeplitz operators"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 709–718</td></tr>
 +
</table>

Latest revision as of 06:44, 15 February 2024

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
How to Cite This Entry:
Toeplitz operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_operator&oldid=13935
This article was adapted from an original article by V.V. Peller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article