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Zapiski Nauchnykh Seminarov LOMI, 1982, Volume 107, Pages 104–135 (Mi znsl3418)  

Sets of simply-invariance

N. G. Makarov
Abstract: Let $X$ be a space of smooth functions on the unit circle $\mathbb T$. Suppose that the operator of multiplication by $z$ is invertible on $X$. A closed set $E$, $E\subset\mathbb T$, is (by definition) the set of simply-invariance for the space $X$ if there exists a function $f$, $f\in X$, such that $f|_E\equiv0$ and $z^{-1}\not\in\operatorname{span}\{z^nf:n\ge0\}$, It is proved that the class of sets of simply-invariance for the spaces $C^n$, $W_p^n$ ($p<\infty$), $\lambda_\omega^n$, coincides with the class of sets of zero Lebesgue measure, for the space $C^\infty$, with the class of Carleson sets, for the space $\Lambda_\omega^n$ with the class of all nowhere dense closed sets. Some related problems are also considered.
English version:
Journal of Soviet Mathematics, 1987, Volume 36, Issue 3, Pages 363–382
DOI: https://doi.org/10.1007/BF01839608
Bibliographic databases:
UDC: 517.5
Language: Russian
Citation: N. G. Makarov, “Sets of simply-invariance”, Investigations on linear operators and function theory. Part X, Zap. Nauchn. Sem. LOMI, 107, "Nauka", Leningrad. Otdel., Leningrad, 1982, 104–135; J. Soviet Math., 36:3 (1987), 363–382
Citation in format AMSBIB
\Bibitem{Mak82}
\by N.~G.~Makarov
\paper Sets of simply-invariance
\inbook Investigations on linear operators and function theory. Part~X
\serial Zap. Nauchn. Sem. LOMI
\yr 1982
\vol 107
\pages 104--135
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3418}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=676152}
\zmath{https://zbmath.org/?q=an:0514.47022|0613.47026}
\transl
\jour J. Soviet Math.
\yr 1987
\vol 36
\issue 3
\pages 363--382
\crossref{https://doi.org/10.1007/BF01839608}
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