# 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 (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 was started by Z. Nehari [a14] and P. Hartman [a9] (cf. also Hardy classes). The following boundedness criterion was proved in [a14]: A matrix determines a bounded operator on if and only if there exists a bounded function on the unit circle such that , , where is the sequence of Fourier coefficients of (cf. also Fourier series). Moreover, the norm of the operator with matrix is equal to

The following compactness criterion was obtained in [a9]: The operator with matrix is compact (cf. also Compact operator) if and only if , , for some continuous function on .

Later it became possible to state these boundedness and compactness criteria in terms of the spaces and . The space of functions of bounded mean oscillation consists of functions such that

where the supremum is taken over all intervals of , is the Lebesgue measure of , and . The space of functions of vanishing mean oscillation consists of functions such that

A combination of the Nehari and Fefferman theorems (see [a6]) gives the following boundedness criterion: The matrix determines a bounded operator on if and only if the function on belongs to . Similarly, the matrix determines a compact operator if and only if .

It is convenient to use different realizations of Hankel operators. The following realization is very important in function theory. Given a function , one defines the Hankel operator by . Here, and is the orthogonal projection onto . A function is called a symbol of (the operator has infinitely many different symbols: for ). The operator has Hankel matrix in the orthonormal basis of and the orthonormal basis of . By Hartman's theorem above, is compact if and only if where is the closed subalgebra of consisting of the functions of the form with and a continuous function on .

For , there exists a function such that ; it is called a best approximation of by analytic functions in the -norm. In general, such a function is not unique (see [a10]). However, if the essential norm (i.e., the distance to the set of compact operators) of is less than its norm, then there is a unique best approximation and the function has constant modulus [a1]. Let . In [a2] it is shown that if the set contains at least two different functions, then this set contains a function of constant modulus ; 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 has finite rank if and only if is a rational function. Moreover, .

Recall that for a bounded linear operator on a Hilbert space, the singular values are defined by

(a1) |

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

Recall that an operator on a Hilbert space belongs to the Schatten–von Neumann class , , if the sequence of its singular values belongs to . The following theorem was obtained in [a16] for and in [a17] and [a23] for : The Hankel operator belongs to if and only if belongs to the Besov space .

There are many different equivalent definitions of Besov spaces. Let . The function belongs to and can be considered as a function analytic in the unit disc . Then if and only if

where is an integer such that and stands for planar Lebesgue measure.

This theorem has many applications, e.g. to rational approximation. For a function on in one can define the numbers by

where is the set of rational functions of degree at most with poles outside .

The following theorem is true: Let and . Then if and only if .

This theorem was obtained in [a16] for , and in [a17], [a15], and [a23] for .

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

For a function one denotes by the unique function satisfying . In [a19], Hankel operators were used to find three big classes of function spaces such that . The first class contains the space and the Besov spaces , . The second class consists of Banach algebras of functions on such that

the trigonometric polynomials are dense in , and the maximal ideal space of can be identified naturally with . The space of functions with absolutely converging Fourier series, the Besov classes , , , 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 for which 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 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 one denotes by the Hankel operator on with Hankel matrix . 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 for piecewise-continuous functions (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 be a self-adjoint operator on a Hilbert space. One can associate with its scalar spectral measure and its spectral multiplicity function (cf. also Spectral function). The following assertion holds: is unitarily equivalent to a Hankel operator if and only if the following conditions are satisfied:

i) is non-invertible;

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

iii) -almost everywhere and -almost everywhere, where is the singular component of .

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

[a1] | V.M. Adamyan, D.Z. Arov, M.G. Krein, "On infinite Hankel matrices and generalized problems of Carathéodory–Fejér and F. Riesz" Funct. Anal. Appl. , 2 (1968) pp. 1–18 Funktsional. Anal. Prilozh. , 2 : 1 (1968) pp. 1–19 |

[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 |

[a4] | V.M. Adamyan, D.Z. Arov, M.G. Krein, "Infinite Hankel block matrices and some related continuation problems" Izv. Akad. Nauk Armyan. SSR Ser. Mat. , 6 (1971) pp. 87–112 |

[a5] | B.A. Francis, "A course in control theory" , Lecture Notes Control and Information Sci. , 88 , Springer (1986) |

[a6] | J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981) |

[a7] | H. Hamburger, "Über eine Erweiterung des Stieltiesschen Momentproblems" Math. Ann. , 81 (1920/1) |

[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 |

[a10] | S. Khavinson, "On some extremal problems of the theory of analytic functions" Transl. Amer. Math. Soc. , 32 : 2 (1963) pp. 139–154 Uchen. Zap. Mosk. Univ. Mat. , 144 : 4 (1951) pp. 133–143 |

[a11] | L. Kronecker, "Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gleichungen" Monatsber. K. Preuss. Akad. Wiss. Berlin (1881) pp. 535–600 |

[a12] | A.V. Megretskii, "A quasinilpotent Hankel operator" Leningrad Math. J. , 2 (1991) pp. 879–889 |

[a13] | A.V. Megretskii, V.V. Peller, S.R. Treil, "The inverse spectral problem for self-adjoint Hankel operators" Acta Math. , 174 (1995) pp. 241–309 |

[a14] | Z. Nehari, "On bounded bilinear forms" Ann. of Math. , 65 (1957) pp. 153–162 |

[a15] | A.A. Pekarskii, "Classes of analytic functions defined by best rational approximations in " Math. USSR Sb. , 55 (1986) pp. 1–18 Mat. Sb. , 127 (1985) pp. 3–20 |

[a16] | V.V. Peller, "Hankel operators of class 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 for , 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 |

[a21] | V.V. Peller, N.J. Young, "Superoptimal singular values and indices of matrix functions" Integral Eq. Operator Th. , 20 (1994) pp. 35–363 |

[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 |

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Hankel operator.

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