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Zap. Nauchn. Sem. POMI, 2012, Volume 401, Pages 71–81 (Mi znsl5226)  

On control subspaces of minimal dimension

M. F. Gamal'

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: The quantity “$\operatorname{disc}$” for a (bounded linear) operator was introduced by N. K. Nikol'skii and V. I. Vasjunin, namely,
$$ \operatorname{disc}T=\sup_{E\in\mathcal R(T)}\min\{\dim E'\colon E'\subset E, E'\in\mathcal R(T)\}, $$
where $\mathcal R(T)$ is the family of all finite dimensional reproducing subspaces for an operator $T$. We give sufficient conditions on operators $T$ under which $\operatorname{disc}T=\infty$. In particular, we show that there exists an operator $T$ with $\operatorname{disc}T=\infty$ and such that $T$ can be represented in the form $T=T_1\oplus T_2$ with $\operatorname{disc}T_1=\operatorname{disc}T_2=1$.

Key words and phrases: normal operator, invariant subspaces.

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English version:
Journal of Mathematical Sciences (New York), 2013, 194:6, 639–644

Bibliographic databases:

UDC: 517.983.243
Received: 08.06.2012

Citation: M. F. Gamal', “On control subspaces of minimal dimension”, Investigations on linear operators and function theory. Part 40, Zap. Nauchn. Sem. POMI, 401, POMI, St. Petersburg, 2012, 71–81; J. Math. Sci. (N. Y.), 194:6 (2013), 639–644

Citation in format AMSBIB
\Bibitem{Gam12}
\by M.~F.~Gamal'
\paper On control subspaces of minimal dimension
\inbook Investigations on linear operators and function theory. Part~40
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 401
\pages 71--81
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5226}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2981967}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2013
\vol 194
\issue 6
\pages 639--644
\crossref{https://doi.org/10.1007/s10958-013-1552-x}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84898975770}


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