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absolutely $p$-summing; bilinear integration; semivariation; tensor product
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.
[1] Bartle R.: A general bilinear vector integral. Studia Math. 15 (1956), 337-351. MR 0080721 | Zbl 0070.28102
[2] Defand A., Floret K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam, 1993. MR 1209438
[3] Diestel J.: Sequences and Series in Banach Spaces. Springer, New York, 1984. MR 0737004
[4] Diestel J., Uhl J.J., Jr.: Vector Measures. Math. Surveys No. 15, Amer. Math. Soc., Providence, 1977. MR 0453964 | Zbl 0521.46035
[5] Diestel J., Jarchow H., Tonge A.: Absolutely Summing Operators. Cambridge University Press, Cambridge, 1995. MR 1342297 | Zbl 1139.47021
[6] Dobrakov I.: On integration in Banach spaces, I. Czechoslovak Math. J. 20 (1970), 511-536. MR 0365138 | Zbl 0215.20103
[7] Dobrakov I.: On integration in Banach spaces, II. Czechoslovak Math. J. 20 (1970), 680-695. MR 0365139 | Zbl 0224.46050
[8] Jefferies B.R.F.: Evolution Processes and the Feynman-Kac Formula. Kluwer Academic Publishers, Dordrecht/Boston/London, 1996. MR 1377058 | Zbl 0844.60027
[9] Jefferies B., Okada S.: Bilinear integration in tensor products. Rocky Mountain J. Math. 28 2 (1998), 517-545. MR 1651584 | Zbl 0936.46035
[10] Kwapien S.: On a theorem of L. Schwartz and its application to absolutely summing operators. Studia Math. 38 (1970), 193-201. MR 0278090
[11] Lindenstrauss J., Tzafriri L.: Classical Banach spaces I. Sequence Spaces. Springer, Berlin, New York, 1977. MR 0500056 | Zbl 0362.46013
[12] Schaefer H.H., Zhang X.-D.: A note on bounded vector measures. Arch. Math. 23 (1994), 152-157. MR 1289297
[13] Swarz C.: Integrating bounded functions for the Dobrakov integral. Math. Slovaca 33 (1983), 141-144. MR 0699082
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