Source: Abstr. Appl. Anal.
(2008), Article ID
651294, 11 pages.
Using some techniques from vector integration,
we prove the weak measurability of the adjoint of strongly
continuous semigroups which factor through Banach spaces
without isomorphic copy of
we also prove the strong continuity away from zero of the
adjoint if the semigroup factors through Grothendieck spaces.
These results are used, in particular, to characterize the
space of strong continuity of
which, in addition, is also characterized for abstract
$L$- and $M$-spaces. As a
corollary, it is proven that abstract
$L$-spaces with no copy of
R. S. Phillips, “The adjoint semigroup,” Pacific Journal of Mathematics, vol. 5, pp. 269–283, 1955.
Mathematical Reviews (MathSciNet): MR70976
J. van Neerven, The Adjoint of a Semigroup of Linear Operators, vol. 1529 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1992.
J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, RI, USA, 1977.
K. Musiał, “Pettis integral,” in Handbook of Measure Theory, Vol. I, II, E. Pap, Ed., chapter 12, pp. 531–586, North-Holland, Amsterdam, The Netherlands, 2002.
K. Musiał, “Topics in the theory of Pettis integration,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 23, no. 1, pp. 177–262, 1991.
K. Yosida, Functional Analysis, vol. 123 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 4th edition, 1974.
D. Bàrcenas and J. Diestel, “Constrained controllability in nonreflexive Banach spaces,” Quaestiones Mathematicae, vol. 18, no. 1–3, pp. 185–198, 1995.
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, “Factoring weakly compact operators,” Journal of Functional Analysis, vol. 17, pp. 311–327, 1974.
J. Diestel, “Grothendieck spaces and vector measures,” in (Proc. Sympos., Alta, Utah, 1972), Vector and Operator Valued Measures and Applications, pp. 97–108, Academic Press, New York, NY, USA, 1973.
J. Diestel, A Survey of Results Related to the Dunford-Pettis Property, vol. 2 of Contemporary Mathematics, American Mathematical Society, Providence, RI, USA, 1980.
Mathematical Reviews (MathSciNet): MR621850
H. P. Lotz, “Uniform convergence of operators on $L^\infty$ and similar spaces,” Mathematische Zeitschrift, vol. 190, no. 2, pp. 207–220, 1985.
Mathematical Reviews (MathSciNet): MR797538
R. Nagel, Ed., One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.
Mathematical Reviews (MathSciNet): MR839450
T. H. Kuo, “On conjugate Banach spaces with the Radon-Nikodym property,” Pacific Journal of Mathematics, vol. 59, no. 2, pp. 497–503, 1975.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Sequence Spaces, Springer, Berlin, Germany, 1977.
L. H. Riddle, E. Saab, and J. J. Uhl Jr., “Sets with the weak Radon-Nikodym property in dual Banach spaces,” Indiana University Mathematics Journal, vol. 32, no. 4, pp. 527–541, 1983.
Mathematical Reviews (MathSciNet): MR703283
E. Odell and H. P. Rosenthal, “A double-dual characterization of separable Banach spaces containing $l^1$,” Israel Journal of Mathematics, vol. 20, no. 3–4, pp. 375–384, 1975.
J. Diestel, Sequences and Series in Banach Spaces, vol. 92 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1984.
Mathematical Reviews (MathSciNet): MR737004
D. Leung, “Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property,” Mathematische Zeitschrift, vol. 197, no. 1, pp. 21–32, 1988.
Mathematical Reviews (MathSciNet): MR917848