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A Generalization of Hankel Operators

✍ Scribed by Rubén A. Martı́nez-Avendaño

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
201 KB
Volume
190
Category
Article
ISSN
0022-1236

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✦ Synopsis

We introduce a class of operators, called l-Hankel operators, as those that satisfy the operator equation S g X -XS=lX, where S is the unilateral forward shift and l is a complex number. We investigate some of the properties of l-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of l-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to l-Hankel operators for certain l. We also study some other operator equations involving S.

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