Hankel Operators on Hilbert Space

Abstract

A famous inequality of D. Hilbert [70], [36] asserts that the matrix commonly known as Hilbert's matrix, determines a bounded linear operator on the Hilbert space of square summable complex sequences. Infinite matrices which possess a similar form to H, namely those that are 'one way infinite' and have identical entries in cross diagonals, are called Hankel matrices, and when these matrices determine bounded operators we have Hankel operators, the subject of this article.The formal companions to the Hankel operators are the Toeplitz operators which have representing matrices possessing a constancy along the long diagonals. Since the stimulating paper of A. Brown and P. R. Halmos [9], the properties of these operators have been extensively developed, culminating in a substantial and sophisticated theory for their spectral, algebraic and C*-algebraic aspects. See, for example, [22], [23] and [64; Chapter 10]. The fact that Hankel operators have not enjoyed such attention is partly due to their dearth of algebraic properties, to the rather mysterious relationship that exists between a Hankel operator and its defining symbol function and to the fact that, in some senses, they are not so natural. . However, there are excellent reasons for studying them, over and above the fact that they form a new and curious class for the attention of operator theorists, and perhaps we should affirm some of these reasons in order to provide an outlook for the reader.It would be desirable to have a context for Hilbert's operator H and to be able to perceive its properties (bounded, not compact, positive, spectrum equal to [0,7r] etc.) as instances of more general theorems concerning the Hankel form.Hankel operators are not as special as one might initially think, at least if one allows unitary equivalence. An integral operator on L 2 (0, oo) whose kernel is of the form k{x + y) is equivalent to a Hankel operator. A prime example is the singular integral operator considered by T. Carleman [12], and which may be viewed as the continuous variant of H. This operator is also the square of the Laplace transform, considered as an

Spectral and scattering theory of fourth order differential operators

“…A proof of its boundedness can be found, e.g., in [12]. We note also that boundedness of the Fourier transform and of the operator T imply that relations (5.42) and (5.43) remain true for all u ∈ L 2 (R + ).…”

Book Review: Hankel operators and their applications

“…The book is devoted to the results of investigations in which the author was involved, and it contains also a good deal of attendant material, starting with elementary facts about Hankel operators. Similar topics were touched upon in some earlier monographs (see, e.g., [3,4,5]), but with lesser thoroughness and completeness. Thus, the appearance of a careful and systematic presentation in one volume is useful and timely, especially if we keep in mind that some proofs are simplified compared to original publications, details are polished, and many quite recent developments are covered.…”

Hankel operators over complex manifolds

“…The traditional concepts of Hankel matrices and Hankel operators on the Hardy space [Pa,Po1,Po2,Zh] have been generalized in various directions [P,Ro1,JPR,Ja2,HR]. Our notion of Hankel forms and (small) Hankel operators essentially coincides with the one in [JPR]; one only has to replace their domain Ω ⊂ C d by M .…”