×

A bilinear version of Orlicz-Pettis theorem. (English) Zbl 1161.46005

The authors establish an Orlicz-Pettis theorem with respect to a bounded, bilinear map. Let \(X,Y,Z\) be Banach spaces and \(b:X\times Y^{\prime }\rightarrow Z\) be a bounded bilinear map. A sequence \( \{x_{n}\}\subset X\) is \(b\)-unconditionally summable if \(\sum_{n=1}^{\infty }\left| \left\langle z^{\prime },b(x_{n},y^{\prime })\right\rangle \right| <\infty \) and for every \(M\subset\mathbb{N}\) there exists \(x_{M}\in X\) such that \(\sum_{n=1}^{\infty }\left\langle z^{\prime },b(x_{n},y^{\prime })\right\rangle =\left\langle z^{\prime },b(x_{M},y^{\prime })\right\rangle \) for all \(z^{\prime }\in Z^{\prime },y^{\prime }\in Y^{\prime }\). The space \(X\) is \(b\)-normed if there exists \(k\) such that \(\left\| x\right\| \leq k\left\| b(x,\cdot )\right\| \) for all \(x\in X\).
The authors show that if the conditions (a) \(X\) is \(b\)-normed, (b) the unit ball of \(Y^{\prime }\) is sequentially compact, and (c) \( b(x,\cdot ):Y^{\prime }\rightarrow Z\) is weak*-norm continuous for every \( x\in X\) are satisfied, then every \(b\)-unconditionally summable sequence is unconditionally summable in \(X\). Applications to the Pettis integral and integration with respect to a vector measure are given. It would be of interest to consider the necessity of the conditions (a)–(c).
The authors also consider spaces of vector valued sequences which are summable with respect to a bounded, bilinear operator. Numerous examples are given.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Blasco, O.; Calabuig, J. M., Hölder inequality for functions that are integrable with respect to bilinear maps, Math. Scand., 102, 1, 101-110 (2008) · Zbl 1161.46023
[2] O. Blasco, J.M. Calabuig, Vector valued functions integrable with respect to bilinear maps, Taiwanese J. Math., in press; O. Blasco, J.M. Calabuig, Vector valued functions integrable with respect to bilinear maps, Taiwanese J. Math., in press · Zbl 1171.42010
[3] O. Blasco, J.M. Calabuig, Fourier analysis with respect to bilinear maps, Acta Math. Sinica, in press; O. Blasco, J.M. Calabuig, Fourier analysis with respect to bilinear maps, Acta Math. Sinica, in press · Zbl 1173.42316
[4] Blasco, O.; Signes, T., Some classes of \(p\)-summing type operators, Bol. Soc. Mat. Mexicana (3), 9, 1, 119-133 (2003), MR1988593 (2004b:47025) · Zbl 1073.47026
[5] Del Campo, R.; Fernández, A.; Ferrando, I.; Mayoral, I.; Naranjo, F., Multiplication operators on spaces of integrable functions with respect to a vector measure, J. Math. Anal. Appl., 343, 1, 514-524 (2008) · Zbl 1154.47028
[6] Diestel, J., Sequences and Series in Banach Spaces, Grad. Texts in Math., vol. 92 (1984), Springer-Verlag: Springer-Verlag New York, xii+261 pp. MR0737004 (85i:46020)
[7] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43 (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, xvi+474 pp. MR1342297 (96i:46001) · Zbl 0855.47016
[8] Diestel, J.; Uhl, J. J., Vector Measures. With a Foreword by B.J. Pettis, Math. Surveys, vol. 15 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, xiii+322 pp. MR0453964 (56 #12216)
[9] Fernández, A.; Mayoral, F.; Naranjo, F.; Sáez, C.; Sánchez Pérez, E. A., Spaces of \(p\)-integrable functions with respect to a vector measure, Positivity, 10, 1, 1-16 (2006), MR2223581 (2006m:46053) · Zbl 1111.46018
[10] Mohsen, A., \(Weak^∗\)-norm sequentially continuous operators, Math. Slovaca, 50, 3, 357-363 (2000), MR1775307 (2003c:46007) · Zbl 0992.46004
[11] Okada, S.; Ricker, W.; Sánchez Pérez, E. A., Optimal Domain and Integral Extension of Operators Acting in Function Spaces, vol. 180 (2008), Birkhäuser, xii+400 pp
[12] Sánchez Pérez, E. A., Compactness arguments for spaces of \(p\)-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces, Illinois J. Math., 45, 3, 907-923 (2001), MR1879243 (2003d:46055) · Zbl 0992.46035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.