On the adjoint of a strongly continuous semigroup. (English) Zbl 1165.47026

In this nicely written paper, the authors prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy of \(l_1\). They obtain the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces.
As consequences of the main results, the authors characterize the strong continuity of \(\{T^{**}(t)\}_{t\geq 0}\), which is also described for abstract \(L\)- and \(M\)-spaces. As a corollary, they prove that abstract \(L\)-spaces with no copy of \(l_1\) are finite-dimensional.


47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI EuDML


[1] R. S. Phillips, “The adjoint semigroup,” Pacific Journal of Mathematics, vol. 5, pp. 269-283, 1955. · Zbl 0064.11202 · doi:10.2140/pjm.1955.5.269
[2] J. van Neerven, The Adjoint of a Semigroup of Linear Operators, vol. 1529 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1992. · Zbl 0780.47026
[3] J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, RI, USA, 1977. · Zbl 0369.46039
[4] K. Musiał, “Pettis integral,” in Handbook of Measure Theory, Vol. I, II, E. Pap, Ed., pp. 531-586, North-Holland, Amsterdam, The Netherlands, 2002, chapter 12. · Zbl 1043.28010
[5] 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. · Zbl 0798.46042
[6] K. Yosida, Functional Analysis, vol. 123 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 4th edition, 1974. · Zbl 0286.46002
[7] D. Bàrcenas and J. Diestel, “Constrained controllability in nonreflexive Banach spaces,” Quaestiones Mathematicae, vol. 18, no. 1-3, pp. 185-198, 1995. · Zbl 0827.46004 · doi:10.1080/16073606.1995.9631794
[8] 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. · Zbl 0306.46020 · doi:10.1016/0022-1236(74)90044-5
[9] 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. · Zbl 0316.46009
[10] J. Diestel, A Survey of Results Related to the Dunford-Pettis Property, vol. 2 of Contemporary Mathematics, American Mathematical Society, Providence, RI, USA, 1980. · Zbl 0571.46013
[11] H. P. Lotz, “Uniform convergence of operators on L\infty and similar spaces,” Mathematische Zeitschrift, vol. 190, no. 2, pp. 207-220, 1985. · Zbl 0623.47033 · doi:10.1007/BF01160459
[12] R. Nagel, Ed., One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986. · Zbl 0585.47030 · doi:10.1007/BFb0074922
[13] T. H. Kuo, “On conjugate Banach spaces with the Radon-Nikodym property,” Pacific Journal of Mathematics, vol. 59, no. 2, pp. 497-503, 1975. · Zbl 0296.46015 · doi:10.2140/pjm.1975.59.497
[14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Sequence Spaces, Springer, Berlin, Germany, 1977. · Zbl 0362.46013
[15] 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. · Zbl 0547.46009 · doi:10.1512/iumj.1983.32.32038
[16] E. Odell and H. P. Rosenthal, “A double-dual characterization of separable Banach spaces containing l1,” Israel Journal of Mathematics, vol. 20, no. 3-4, pp. 375-384, 1975. · Zbl 0312.46031 · doi:10.1007/BF02760341
[17] J. Diestel, Sequences and Series in Banach Spaces, vol. 92 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1984. · Zbl 0542.46007
[18] D. Leung, “Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property,” Mathematische Zeitschrift, vol. 197, no. 1, pp. 21-32, 1988. · Zbl 0618.46023 · doi:10.1007/BF01161628
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.