# Article

 Title: Semivariation in $L^p$-spaces (English) Author: Jefferies, Brian Author: Okada, Susumu Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 46 Issue: 3 Year: 2005 Pages: 425-436 . Category: math . Summary: Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X\hat\otimes_\tau Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X\hat\otimes_\tau Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck's theorem may be applied. In this paper, we take an index $1 \le p < \infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with $\tau_p$ the associated $L^p$-tensor product topology. An application of Orlicz's lemma shows that not all uniformly bounded $X$-valued functions are integrable in $X\hat\otimes_{\tau_p} Y$ with respect to a $Y$-valued measure in the case $1\le p < 2$. For $2 < p <\infty$, the negative result is equivalent to the fact that not all continuous linear maps from $\ell^1$ to $\ell^p$ are $p$-summing, which follows from a result of S. Kwapien. (English) Keyword: absolutely $p$-summing Keyword: bilinear integration Keyword: semivariation Keyword: tensor product MSC: 28B05 MSC: 46B42 MSC: 46G10 MSC: 47B65 idZBL: Zbl 1121.28013 idMR: MR2174522 . Date available: 2009-05-05T16:52:14Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119538 . Reference: [1] Bartle R.: A general bilinear vector integral.Studia Math. 15 (1956), 337-351. Zbl 0070.28102, MR 0080721 Reference: [2] Defand A., Floret K.: Tensor Norms and Operator Ideals.North-Holland, Amsterdam, 1993. MR 1209438 Reference: [3] Diestel J.: Sequences and Series in Banach Spaces.Springer, New York, 1984. MR 0737004 Reference: [4] Diestel J., Uhl J.J., Jr.: Vector Measures.Math. Surveys No. 15, Amer. Math. Soc., Providence, 1977. Zbl 0521.46035, MR 0453964 Reference: [5] Diestel J., Jarchow H., Tonge A.: Absolutely Summing Operators.Cambridge University Press, Cambridge, 1995. Zbl 1139.47021, MR 1342297 Reference: [6] Dobrakov I.: On integration in Banach spaces, I.Czechoslovak Math. J. 20 (1970), 511-536. Zbl 0215.20103, MR 0365138 Reference: [7] Dobrakov I.: On integration in Banach spaces, II.Czechoslovak Math. J. 20 (1970), 680-695. Zbl 0224.46050, MR 0365139 Reference: [8] Jefferies B.R.F.: Evolution Processes and the Feynman-Kac Formula.Kluwer Academic Publishers, Dordrecht/Boston/London, 1996. Zbl 0844.60027, MR 1377058 Reference: [9] Jefferies B., Okada S.: Bilinear integration in tensor products.Rocky Mountain J. Math. 28 2 (1998), 517-545. Zbl 0936.46035, MR 1651584 Reference: [10] Kwapien S.: On a theorem of L. Schwartz and its application to absolutely summing operators.Studia Math. 38 (1970), 193-201. MR 0278090 Reference: [11] Lindenstrauss J., Tzafriri L.: Classical Banach spaces I. Sequence Spaces.Springer, Berlin, New York, 1977. Zbl 0362.46013, MR 0500056 Reference: [12] Schaefer H.H., Zhang X.-D.: A note on bounded vector measures.Arch. Math. 23 (1994), 152-157. MR 1289297 Reference: [13] Swarz C.: Integrating bounded functions for the Dobrakov integral.Math. Slovaca 33 (1983), 141-144. MR 0699082 .

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