Hankel operator

The Hankel operators form a class of operators which is one of the most important classes of operators in function theory; it has many applications in different fields of mathematics and applied mathematics.

A Hankel operator can be defined as an operator whose matrix has the form $( \alpha _ { j + k} ) _ { j , k \geq 0}$ (such matrices are called Hankel matrices, cf. also Padé approximation). Finite matrices whose entries depend only on the sum of the coordinates were studied first by H. Hankel [a8]. One of the first results on infinite Hankel matrices was obtained by L. Kronecker [a11], who described the finite-rank Hankel matrices. Hankel operators played an important role in moment problems [a8] as well as in other classical problems of analysis.

The study of Hankel operators on the Hardy class $H ^ { 2 }$ was started by Z. Nehari [a14] and P. Hartman [a9] (cf. also Hardy classes). The following boundedness criterion was proved in [a14]: A matrix $( \alpha _ { j + k} ) _ { j , k \geq 0}$ determines a bounded operator on $\text{l} ^ { 2 }$ if and only if there exists a bounded function $\phi$ on the unit circle $\bf T$ such that $\widehat { \phi } ( j ) = \alpha_j$, $j \geq 0$, where $\{ \hat { \phi } ( j ) \} _ { j \geq 0 }$ is the sequence of Fourier coefficients of $\phi$ (cf. also Fourier series). Moreover, the norm of the operator with matrix $( \alpha _ { j + k} ) _ { j , k \geq 0}$ is equal to

\begin{equation*} \operatorname { inf } \left\{ \| \phi \| _ { \infty } : \phi \in L ^ { \infty } , \widehat { \phi } ( j ) = \alpha _ { j } \text { for } j \geq 0 \right\}. \end{equation*}

The following compactness criterion was obtained in [a9]: The operator with matrix $( \alpha _ { j + k} ) _ { j , k \geq 0}$ is compact (cf. also Compact operator) if and only if $\alpha_{ j} = \widehat { \phi } ( j )$, $j \geq 0$, for some continuous function $\phi$ on $\bf T$.

Later it became possible to state these boundedness and compactness criteria in terms of the spaces $\operatorname{BMO}$ and $\operatorname{VMO}$. The space $\operatorname{BMO}$ of functions of bounded mean oscillation consists of functions $f \in L ^ { 1 } ( \mathbf{T} )$ such that

\begin{equation*} \operatorname { sup } _ { I } \frac { 1 } { | I | } \int _ { I } | f - f _ { I } | d m < \infty, \end{equation*}

where the supremum is taken over all intervals $I$ of $\bf T$, $| l | = m ( l )$ is the Lebesgue measure of $I$, and $f _ { I } = ( 1 / | I | ) \int _ { I } f d m$. The space $\operatorname{VMO}$ of functions of vanishing mean oscillation consists of functions $f \in L ^ { 1 } ( \mathbf{T} )$ such that

\begin{equation*} \operatorname { lim } _ { | I | \rightarrow 0 } \frac { 1 } { | I | } \int _ { I } | f - f _ { I } | d m = 0. \end{equation*}

Cf. also $\operatorname{BMO}$-space; $\operatorname{VMO}$-space.

A combination of the Nehari and Fefferman theorems (see [a6]) gives the following boundedness criterion: The matrix $( \alpha _ { j + k} ) _ { j , k \geq 0}$ determines a bounded operator on $\text{l} ^ { 2 }$ if and only if the function $\sum _ { j \geq 0 } \alpha _ { j } z ^ { j }$ on $\bf T$ belongs to $\operatorname{BMO}$. Similarly, the matrix $( \alpha _ { j + k} ) _ { j , k \geq 0}$ determines a compact operator if and only if $\sum _ { j \geq 0 } \alpha _ { j } z ^ { j } \in \operatorname{VMO}$.

It is convenient to use different realizations of Hankel operators. The following realization is very important in function theory. Given a function $\phi \in L ^ { \infty }$, one defines the Hankel operator $H _ { \phi } : H ^ { 2 } \rightarrow H _ { - } ^ { 2 }$ by $H _ { \phi } f = \mathcal{P} _ { - } \phi f$. Here, $H_- ^ { 2 } = L ^ { 2 } \ominus H ^ { 2 }$ and $\mathcal{P}_ {-}$ is the orthogonal projection onto $H_{-} ^ { 2 }$. A function $\phi$ is called a symbol of $H _ { \phi }$ (the operator $H _ { \phi }$ has infinitely many different symbols: $H _ { \phi } = H _ { \phi + \psi }$ for $\psi \in H ^ { \infty }$). The operator $H _ { \phi }$ has Hankel matrix $( \hat { \phi } ( - j - k - 1 ) )_{ j > 0 , k \geq 0}$ in the orthonormal basis $\{ z ^ { k } \} _ { k \geq 0 }$ of $H ^ { 2 }$ and the orthonormal basis $\{ \overline{z} \square ^ { j } \}_{j > 0}$ of $H_{-} ^ { 2 }$. By Hartman's theorem above, $H _ { \phi }$ is compact if and only if $\phi \in H ^ { \infty } + C$ where $H ^ { \infty } + C$ is the closed subalgebra of $L^{\infty}$ consisting of the functions of the form $f + g$ with $f \in H ^ { \infty }$ and $g$ a continuous function on $\bf T$.

For $\phi \in L ^ { \infty }$, there exists a function $f \in H ^ { \infty }$ such that $\| \phi - f \| _ { L^\infty } = \| H _ { \phi } \|$; it is called a best approximation of $\phi$ by analytic functions in the $L^{\infty}$-norm. In general, such a function $f$ is not unique (see [a10]). However, if the essential norm (i.e., the distance to the set of compact operators) of $H _ { \phi }$ is less than its norm, then there is a unique best approximation $\phi$ and the function $\phi - f$ has constant modulus [a1]. Let $\rho \geq \| H _ { \phi } \|$. In [a2] it is shown that if the set $\{ f \in H ^ { \infty } : \| \phi - f \| _ { L } \infty \leq \rho \}$ contains at least two different functions, then this set contains a function of constant modulus $\rho$; a formula which parameterizes all functions in this set has also been obtained [a2].

A description of the Hankel operators of finite rank was given in [a11]: The Hankel operator $H _ { \phi }$ has finite rank if and only if $\mathcal{P} - \phi$ is a rational function. Moreover, $\operatorname{rank} H _ { \phi } = \operatorname { deg } {\cal P}_{-} \phi$.

Recall that for a bounded linear operator $T$ on a Hilbert space, the singular values $s _ { j } ( T )$ are defined by

\begin{equation} \tag{a1} s _ { j } ( T ) = \operatorname { inf } \{ \| T - R \| : \operatorname { rank } R \leq j \} , j \geq 0. \end{equation}

In [a3] the following, very deep, theorem was obtained: If $T$ is a Hankel operator, then in (a1) it is sufficient to consider only Hankel operators $R$ of rank at most $j$.

Recall that an operator $T$ on a Hilbert space belongs to the Schatten–von Neumann class $\mathcal{S} _ { p }$, $0 < p < \infty$, if the sequence $\{ s _ { j } ( T ) \} _ { j \geq 0 }$ of its singular values belongs to $\mathbf{l}^{p}$. The following theorem was obtained in [a16] for $1 \leq p < \infty$ and in [a17] and [a23] for $0 < p < 1$: The Hankel operator $H _ { \phi }$ belongs to $\mathcal{S} _ { p }$ if and only if $\mathcal{P} - \phi$ belongs to the Besov space $B _ { p } ^ { 1 / p }$.

There are many different equivalent definitions of Besov spaces. Let $\psi = \overline { \mathcal{P} - \phi }$. The function $\psi$ belongs to $H ^ { 2 }$ and can be considered as a function analytic in the unit disc $D$. Then $\mathcal{P} _ { - } \phi \in B _ { p } ^ { 1 / p }$ if and only if

\begin{equation*} \int _ { D } | \psi ^ { ( n ) } ( \zeta ) | ^ { p } ( 1 - | \zeta | ) ^ { n p - 2 } d m _ { 2 } ( \zeta ) < \infty, \end{equation*}

where $n$ is an integer such that $n > 1 / p$ and $m _ { 2 }$ stands for planar Lebesgue measure.

This theorem has many applications, e.g. to rational approximation. For a function $\phi$ on $\bf T$ in $\operatorname{BMO}$ one can define the numbers $\rho _ { n } ( \phi )$ by

\begin{equation*} \rho _ { n } ( \phi ) = \operatorname { inf } \{ \| \phi - r \| _ { \operatorname{BMO} } : \rho \in \mathcal{R} _ { n } \}, \end{equation*}

where $\mathcal{R} _ { n }$ is the set of rational functions of degree at most $n$ with poles outside $\bf T$.

The following theorem is true: Let $\phi \in \operatorname{BMO}$ and $0 < p < \infty$. Then $\{ \rho _ { n } ( \phi ) \} _ { n \geq 0} \in \text{l} ^ { p }$ if and only if $\phi \in B _ { p } ^ { 1 / p }$.

This theorem was obtained in [a16] for $1 \leq p < \infty$, and in [a17], [a15], and [a23] for $0 < p < 1$.

Among the numerous applications of Hankel operators, heredity results for the non-linear operator $\mathcal{A}$ of best approximation by analytic functions can be found in [a19].

For a function $\phi \in \operatorname{VMO}$ one denotes by $\mathcal{A} \phi$ the unique function $f \in \operatorname{BMOA} = \operatorname{BMO} \cap H ^ { 2 }$ satisfying $\| \phi - f \| _ { L ^{\infty} ( \mathbf{T} )} = \| H _ { \phi } \|$. In [a19], Hankel operators were used to find three big classes of function spaces $X$ such that $\mathcal{A} X \subset X$. The first class contains the space $\operatorname{VMO}$ and the Besov spaces $B _ { p } ^ { 1 / p }$, $0 < p < \infty$. The second class consists of Banach algebras $X$ of functions on $\bf T$ such that

\begin{equation*} f \in X \text{ implies } \bar{f} \in X \text{ and } \mathcal{P}_-f \in X, \end{equation*}

the trigonometric polynomials are dense in $X$, and the maximal ideal space of $X$ can be identified naturally with $\bf T$. The space of functions with absolutely converging Fourier series, the Besov classes $B _ { p } ^ { S }$, $1 \leq p < \infty$, $s > 1 / p$, and many other classical Banach spaces of functions satisfy the above conditions. The third class found in [a19] include non-separable Banach spaces (e.g., Hölder and Zygmund classes) as well as certain locally convex spaces. Note, however, that there are continuous functions $\phi$ for which $\mathcal{A} \phi$ is discontinuous.

Hankel operators were also used in [a19] to obtain many results on regularity conditions for stationary random processes (cf. also Stationary stochastic process).

Hankel operators are very important in systems theory and control theory (see [a5] and also $H ^ { \infty }$ control theory).

Another realization of Hankel operators, as operators on the same Hilbert space, makes it possible to study their spectral properties. For a function $\phi \in L ^ { \infty }$ one denotes by $\Gamma _ { \phi }$ the Hankel operator on $\text{l} ^ { 2 }$ with Hankel matrix $\{ \widehat { \phi } ( j + k ) \}_{ j , k \geq 0}$. It is a very difficult problem to describe the spectral properties of such Hankel operators. Known results include the following ones. S. Power has described the essential spectrum of $\Gamma _ { \phi }$ for piecewise-continuous functions $\phi$ (see [a22]). An example of a non-zero quasi-nilpotent Hankel operator was constructed in [a12].

In [a13], the problem of the spectral characterization of self-adjoint Hankel operators was solved. Let $A$ be a self-adjoint operator on a Hilbert space. One can associate with $A$ its scalar spectral measure $\mu$ and its spectral multiplicity function $\nu$ (cf. also Spectral function). The following assertion holds: $A$ is unitarily equivalent to a Hankel operator if and only if the following conditions are satisfied:

i) $A$ is non-invertible;

ii) the kernel of $A$ is either trivial or infinite-dimensional;

iii) $| \nu ( t ) - \nu ( - t ) | \leq 2$ $\mu$-almost everywhere and $| \nu ( t ) - \nu ( - t ) | \leq 1$ $\mu _ { \text{s} }$-almost everywhere, where $\mu _ { \text{s} }$ is the singular component of $\mu$.

The proof of this result is based on linear dynamical systems.

In applications (such as to prediction theory, control theory, or systems theory) it is important to consider Hankel operators with matrix-valued symbols; see [a4] for the basic properties of such operators. Hankel operators with matrix symbols were used in [a20], [a21] to study approximation problems for matrix-valued functions (so-called superoptimal approximations). See also [a24] for another approach to this problem.

The recent (1998) survey [a18] gives more detailed information on Hankel operators.

Finally, there are many results on analogues of Hankel operators on the unit ball, the poly-disc and many other domains.

References

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[a2]	V.M. Adamyan, D.Z. Arov, M.G. Krein, "On infinite Hankel matrices and generalized problems of Carathéodory–Fejér and I. Schur" Funct. Anal. Appl. , 2 (1968) pp. 269–281 Funktsional. Anal. i Prilozh. , 2 : 2 (1968) pp. 1–17
[a3]	V.M. Adamyan, D.Z. Arov, M.G. Krein, "Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Takagi problem" Math. USSR Sb. , 15 (1971) pp. 31–73 Mat. Sb. , 86 (1971) pp. 34–75
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[a8]	H. Hankel, "Ueber eine besondre Classe der symmetrishchen Determinanten" (Leipziger) Diss. Göttingen (1861)
[a9]	P. Hartman, "On completely continuous Hankel matrices" Proc. Amer. Math. Soc. , 9 (1958) pp. 862–866
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[a12]	A.V. Megretskii, "A quasinilpotent Hankel operator" Leningrad Math. J. , 2 (1991) pp. 879–889
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[a16]	V.V. Peller, "Hankel operators of class $\mathfrak{S}_p$ and applications (rational approximation, Gaussian processes, the majorization problem for operators)" Math. USSR Sb. , 41 (1982) pp. 443–479 Mat Sb. , 113 (1980) pp. 538–581
[a17]	V.V. Peller, "A description of Hankel operators of class $\mathfrak{S}_p$ for $p > 0$, investigation of the rate of rational approximation and other applications" Math. USSR Sb. , 50 (1985) pp. 465–494 Mat. Sb. , 122 (1983) pp. 481–510
[a18]	V.V. Peller, "An excursion into the theory of Hankel operators" , Holomorphic Function Spaces Book. Proc. MSRI Sem. Fall 1995 (1995)
[a19]	V.V. Peller, S.V. Khrushchev, "Hankel operators, best approximation and stationary Gaussian processes" Russian Math. Surveys , 37 : 1 (1982) pp. 61–144 Uspekhi Mat. Nauk , 37 : 1 (1982) pp. 53–124
[a20]	V.V. Peller, N.J. Young, "Superoptimal analytic approximations of matrix functions" J. Funct. Anal. , 120 (1994) pp. 300–343
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[a22]	S. Power, "Hankel operators on Hilbert space" , Pitman (1982)
[a23]	S. Semmes, "Trace ideal criteria for Hankel operators and applications to Besov classes" Integral Eq. Operator Th. , 7 (1984) pp. 241–281
[a24]	S.R. Treil, "On superoptimal approximation by analytic and meromorphic matrix-valued functions" J. Funct. Anal. , 131 (1995) pp. 243–255