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Санкт-Петербургский семинар по теории операторов и теории функций
21 февраля 2022 г. 18:30–20:00, г. Санкт-Петербург, на платформе zoom, пароль можно узнать у Д. Столярова http://www.mathnet.ru/php/person.phtml?option_lang=rus&personid=61744 <http://www.mathnet.ru/php/person.phtml?option_lang=rus&personid=61744>преполагается трансляция в ПОМИ., комната 311
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[Boundary Values in Spaces Spanned by Rational Functions and the Index of Invariant Subspaces]
Дж. Э. Бреннан |
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Аннотация:
Let $\mu$ be a positive compactly supported measure in the complex plane $\mathbb C$, and
for each $p,1\leq p<\infty,$ let $H^p(\mu)$ be the closed subspace of $L^p(\mu)$
spanned by the polynomials. In 1991 Thomson gave a complete description of its
structure, expressing $H^p(\mu)$ as the direct sum of invariant subspaces, all but
one of which is irreducible in the sense that it contains no non-trivial
characteristic function. Years later, Aleman, Richter and Sundberg gave a more
detailed analysis of the invariant subspaces in any irreducible summand. Here we
discuss the extent to which those earlier results can be extended to $R^p(\mu)$, the
closed subspace of $L^p(\mu)$ spanned by the rational functions having no poles on
the support of $\mu$, by first establishing the existence of boundary values in
these spaces. Since the measure $\mu$ may have two-dimensional support we shall make
use of the concept of balayage as employed by Havin in 1965 to study the
boundary properties of Cauchy-type integrals.
Язык доклада: английский
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