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Multiplication operators on spaces of integrable functions with respect to a vector measure. (English) Zbl 1154.47028

The authors study multiplication operators between \(L^p\)-spaces of vector measures, such as the spaces \(L_w^p(m)\) of weakly \(p\)-integrable functions and \(L_p(m)\) of \(p\)-integrable functions with \(1 \leq p \leq \infty\) and a countably additive measure \(m\) defined on a \(\sigma\)-algebra \(\Sigma\) of subsets of some set \(\Omega\) and taking values in a real Banach space \(X\). The majority of the results establish necessary and sufficient conditions on a function \(g \in L^p(m)\), so that the multiplication operator \(M_g f = g \cdot f\) is a bounded linear operator from one of the above spaces to another.

MSC:

47B38 Linear operators on function spaces (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46G10 Vector-valued measures and integration
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References:

[1] Abramovich, Y. A.; Aliprantis, C. D.; Burkinshaw, O., Multiplication and compact-friendly operators, Positivity, 1, 171-180 (1997), MR 1658340 · Zbl 0907.47032
[2] Abramovich, Y. A.; Aliprantis, C. D.; Burkinshaw, O.; Wickstead, A. W., A characterization of compact-friendly multiplication operators, Indag. Math. (N.S.), 10, 161-171 (1997), MR 1816212 · Zbl 0907.47032
[3] Aliprantis, C. D.; Burkinshaw, O., Positive Operators, Pure Appl. Math., vol. 119 (1985), Academic Press: Academic Press Orlando, FL, MR 0809372 · Zbl 0567.47037
[4] Curbera, G. P., Banach space properties of \(L^1\) of a vector measure, Proc. Amer. Math. Soc., 123, 3797-3806 (1995), MR 1285984 · Zbl 0848.46015
[5] Curbera, G. P.; Ricker, W. J., Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.), 17, 187-204 (2006) · Zbl 1106.47026
[6] Curbera, G. P.; Ricker, W. J., The Fatou property in \(p\)-convex Banach lattices, J. Math. Anal. Appl., 328, 287-294 (2007), MR 2285548 · Zbl 1121.46017
[7] Diestel, J.; Uhl, J. J., Vector Measures, Math. Surveys, vol. 15 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, MR 0453964 · Zbl 0369.46039
[8] 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, 1-16 (2006), MR 2223581 · Zbl 1111.46018
[9] Kluvánek, I.; Knowles, G., Vector Measures and Control Systems, Notas Mat., vol. 58 (1975), North-Holland: North-Holland Amsterdam, MR 0499068
[10] Lewis, D. R., Integration with respect to vector measures, Pacific J. Math., 33, 157-165 (1970), MR 0259064 · Zbl 0195.14303
[11] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces II. Function Spaces, Ergeb. Math. Grenzgeb., vol. 97 (1979), Springer-Verlag: Springer-Verlag Berlin, MR 0540367 · Zbl 0403.46022
[12] Meyer-Nieberg, P., Banach Lattices (1991), Springer-Verlag: Springer-Verlag Berlin, MR 1128093 · Zbl 0743.46015
[13] Sirotkin, G. G., Compact-friendly multiplication operators on Banach function spaces, J. Funct. Anal., 192, 517-523 (2002), MR 1923412 · Zbl 1034.47011
[14] Stefansson, G. F., \(L_1\) of a vector measure, Matematiche, 48, 219-234 (1993), MR 1320665 · Zbl 0827.46042
[15] Takagi, H.; Yokouchi, K., Multiplication and composition operators between two \(L^p\)-spaces, Contemp. Math., 232, 321-338 (1999), MR 1678344 · Zbl 0939.47028
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