Boundary-value problems with a shift for analytic functions and singular functional equations

Зверович, Эдмунд Иванович; Litvinchuk, Georgii S.

doi:10.1070/rm1968v023n03abeh003774

Cited by 20 publications

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“…If v is a positive measure on (0, 00) then let Z v be the Laplace transform regarded as a linear transformation from L 2 (0, 00) to L 2 (v). Then Z*Z V is the operator S k on L 2 (0, 00) given by S k f(x) = I k(x + y)f(y)dy (3.8) where k is the Laplace transform of v. Part (a) is stated in H. Widom [73] and the theorem can be proved by establishing an equivalence with the circle situation and the operators S\_n~].…”

Section: -Imentioning

confidence: 99%

“…In this case the shift is the mapping z -> z, on \z\ = 1, which determines J. There is a sizeable literature ( [45], [73], [34], [18] for example) about the spectral, Fredholm and index theory of these operators and of their generalisations, where, in effect, the singular integrals related to P are replaced by singular integrals on simple closed Lyapunov contours in the complex plane. It seems likely that a study of these grand C*-algebras will lead to a unification of the essential spectral aspects of Toeplitz operators, Hankel operators and the Carleman shift operators.…”

Section: S^infhs^-sj (42)mentioning

confidence: 99%

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Hankel Operators on Hilbert Space

Power

1980

Bulletin of the London Mathematical Society

137

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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

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Section: -Imentioning

confidence: 99%

Section: S^infhs^-sj (42)mentioning

confidence: 99%

Hankel Operators on Hilbert Space

Power

1980

Bulletin of the London Mathematical Society

137

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“…The paper [12] in its time was a breakthrough in the theory of singular integral operators with involutive shift. In clarification of the results accumulated earlier (see, i.e., [Lit67,ZL68,Ant70,KS72b,KS72a]), the authors came up with the relation which makes crystal clear the relation between the Fredholm properties and the index of the operator A (with shift) and the associated with it operatorÃ W (without shift but acting on the space of vector functions with the size doubled). Moreover, for the operators with orientation preserving shift the relation actually holds for arbitrary measurable matrix coefficients, not just in the continuous case.…”

Section: Introduction Leonid Lerer Vadim Olshevsky and Ilya Spitkovskymentioning

confidence: 99%