Research Problems in Complex Analysis

“…This way however is not so natural for positive Hankel operators and the problem 'which L°° functions generate positive Hankel operators?' (posed by F. Holland [4;Problem 8.14]) is still open as far as the author is aware.…”

Section: Schmidt Condition Leads Us To Conclude Using Jensen's Inequmentioning

confidence: 99%

Hankel Operators on Hilbert Space

Power

1980

Bulletin of the London Mathematical Society

137

View full text Add to dashboard Cite

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

show abstract

“…Clunie [1] (problem *2.49) posed the question as to whether or not the set M can have isolated points. Following Hardy [2], when points of the set M lie on all or part of curves, we shall define those parts of the curves as maximum curves.…”

Section: Introductionmentioning

confidence: 99%

Maximum curves and isolated points of entire functions

Tyler

2000

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Given M (r, f ) = max |z|=r (|f (z)|) , curves belonging to the set of points M = {z : |f (z)| = M (|z|, f)} were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f 1 (z) and f 2 (z), if the maximum curve of f 1 (z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f 1 (z)f 2 (z). By means of these results an entire function of infinite order is constructed for which the set M has an infinite number of isolated points. A polynomial is also constructed with an isolated point.

show abstract

Research Problems in Complex Analysis

Cited by 51 publications

References 16 publications

Minimal Interpolation by Blaschke Products

Minimal Interpolation by Blaschke Products

Hankel Operators on Hilbert Space

Maximum curves and isolated points of entire functions

Contact Info

Product

Resources

About