Completely continuous multilinear operators on 𝐶(𝐾) spaces

Villanueva, Ignacio

doi:10.1090/S0002-9939-99-05396-4

Completely continuous multilinear operators on $C(K)$ spaces
HTML articles powered by AMS MathViewer

by Ignacio Villanueva PDF

Proc. Amer. Math. Soc. 128 (2000), 793-801 Request permission

Abstract:

Given a $k$-linear operator $T$ from a product of $C(K)$ spaces into a Banach space $X$, our main result proves the equivalence between $T$ being completely continuous, $T$ having an $X$-valued separately $\omega ^*-\omega ^*$ continuous extension to the product of the biduals and $T$ having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to $T$ being weakly compact, and that, for $k>1$, $T$ being weakly compact implies the conditions above but the converse fails.

References

Richard M. Aron and Paul D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3–24 (English, with French summary). MR 508947
R. M. Aron, B. J. Cole, and T. W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93. MR 1096902
Richard M. Aron and Pablo Galindo, Weakly compact multilinear mappings, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 181–192. MR 1437822, DOI 10.1017/S0013091500023543
R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
Fernando Bombal and Pilar Cembranos, Characterization of some classes of operators on spaces of vector-valued continuous functions, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 137–146. MR 764502, DOI 10.1017/S0305004100062678
F. Bombal, and I. Villanueva, Multilinear operators in spaces of continuous functions. Funct. Approx. Comment. Math. XXVI (1998), 117–126.
J. M. F. Castillo and M. Gonzáles, On the Dunford-Pettis property in Banach spaces, Acta Univ. Carolin. Math. Phys. 35 (1994), no. 2, 5–12. MR 1366492
Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 15–60. MR 621850
Ivan Dobrakov, On integration in Banach spaces. VIII. Polymeasures, Czechoslovak Math. J. 37(112) (1987), no. 3, 487–506. MR 904773
Ivan Dobrakov, Representation of multilinear operators on ${\mathsf X}C_0(T_i)$, Czechoslovak Math. J. 39(114) (1989), no. 2, 288–302. MR 992135
Manuel González and Joaquín M. Gutiérrez, When every polynomial is unconditionally converging, Arch. Math. (Basel) 63 (1994), no. 2, 145–151. MR 1289296, DOI 10.1007/BF01189888
Manuel González and Joaquín M. Gutiérrez, Unconditionally converging polynomials on Banach spaces, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 2, 321–331. MR 1307084, DOI 10.1017/S030500410007314X
A. Pełczyński, On weakly compact polynomial operators on $B$-spaces with Dunford-Pettis property, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 371–378. MR 161160
A. Pełczyński, A theorem of Dunford-Pettis type for polynomial operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 379–386. MR 161161
Raymond A. Ryan, Dunford-Pettis properties, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 5, 373–379 (English, with Russian summary). MR 557405
I. Villanueva, Polimedidas y representación de operadores multilineales de $C(\Omega _1, X_1)\times \cdots \times C(\Omega _d, X_d)$. Tesina de Licenciatura, Dpto. de Análisis Matemático, Fac. de Matemáticas, Universidad Complutense de Madrid, (1997).