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 Zap. Nauchn. Sem. POMI, 2005, Volume 327, Pages 5–16 (Mi znsl320)

A version of the Grothendieck theorem for subspaces of analytic functions in lattices

D. S. Anisimov

Saint-Petersburg State University

Abstract: A version of Grothendieck's inequality says that any bounded linear operator acting from a Banach lattice $X$ to a Banach lattice $Y$, also acts from $X(\ell^2)$ to $Y(\ell^2)$. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let $X$ be a Banach lattice of measurable functions on the circle, and let an operator $T$ act from the corresponding subspace of analytic functions $X_A$ to a Banach lattice $Y$ or, if $Y$ is also a lattice of measurable functions on the circle, to the quotient space $Y/Y_A$. Under certain mild conditions on the lattices involved, it is proved that $T$ induces an operator acting from $X_A(\ell^2)$ to $Y(\ell^2)$ or to $Y/Y_A(\ell^2)$, respectively.

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English version:
Journal of Mathematical Sciences (New York), 2006, 139:2, 6363–6368

Bibliographic databases:

UDC: 517.5

Citation: D. S. Anisimov, “A version of the Grothendieck theorem for subspaces of analytic functions in lattices”, Investigations on linear operators and function theory. Part 33, Zap. Nauchn. Sem. POMI, 327, POMI, St. Petersburg, 2005, 5–16; J. Math. Sci. (N. Y.), 139:2 (2006), 6363–6368

Citation in format AMSBIB
\Bibitem{Ani05} \by D.~S.~Anisimov \paper A version of the Grothendieck theorem for subspaces of analytic functions in lattices \inbook Investigations on linear operators and function theory. Part~33 \serial Zap. Nauchn. Sem. POMI \yr 2005 \vol 327 \pages 5--16 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl320} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2184425} \zmath{https://zbmath.org/?q=an:1091.46018} \transl \jour J. Math. Sci. (N. Y.) \yr 2006 \vol 139 \issue 2 \pages 6363--6368 \crossref{https://doi.org/10.1007/s10958-006-0353-x} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33750145429} 

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This publication is cited in the following articles:
1. D. S. Anisimov, S. V. Kislyakov, “Strong factorization of operators defined on subspaces of analytic functions in lattices”, J. Math. Sci. (N. Y.), 141:5 (2007), 1511–1516
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