Characterizations of

weakly compact operators on

Author:
T. V. Panchapagesan

Journal:
Trans. Amer. Math. Soc. **350** (1998), 4849-4867

MSC (1991):
Primary 47B38, 46G10; Secondary 28B05

MathSciNet review:
1615942

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

**[BDS]**R. G. Bartle, N. Dunford, and J. Schwartz,*Weak compactness and vector measures*, Canad. J. Math.**7**(1955), 289–305. MR**0070050****[DU]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[Di]**N. Dinculeanu,*Vector Measures*, Pergamon Press, New York, (1967). MR**34:6011****[DK]**N. Dinculeanu and I. Kluvanek,*On vector measures*, Proc. London Math. Soc. (3)**17**(1967), 505–512. MR**0214722****[DL]**N. Dinculeanu and Paul W. Lewis,*Regularity of Baire measures*, Proc. Amer. Math. Soc.**26**(1970), 92–94. MR**0260968**, 10.1090/S0002-9939-1970-0260968-4**[Do]**Ivan Dobrakov,*On integration in Banach spaces. IV*, Czechoslovak Math. J.**30(105)**(1980), no. 2, 259–279. With a loose Russian summary. MR**566051****[E]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR**0221256****[G]**A. Grothendieck,*Sur les applications linéaires faiblement compactes d’espaces du type 𝐶(𝐾)*, Canadian J. Math.**5**(1953), 129–173 (French). MR**0058866****[Ha]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869****[Ho]**John Horváth,*Topological vector spaces and distributions. Vol. I*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR**0205028****[K]**Igor Kluvánek,*Characterization of Fourier-Stieltjes transforms of vector and operator valued measures*, Czechoslovak Math. J.**17 (92)**(1967), 261–277 (English, with Russian summary). MR**0230872****[M]**H. Reiter,*Subalgebras of 𝐿¹(𝐺)*, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math.**27**(1965), 691–696. MR**0196515****[Pa1]**T.V. Panchapagesan,*On complex Radon measures I*, Czech. Math. J.**42**, (1992), 599-612. MR**94e:28014****[Pa2]**T. V. Panchapagesan,*On complex Radon measures. II*, Czechoslovak Math. J.**43(118)**(1993), no. 1, 65–82. MR**1205231****[Pa3]**T. V. Panchapagesan,*Abstract regularity of additive and 𝜎-additive group-valued set functions*, Math. Slovaca**45**(1995), no. 4, 381–393. MR**1387055****[Pa4]**-,*Baire and -Borel characterizations of weakly compact sets in*, Trans. Amer. Math. Soc.**350**(1998), 4839-4847.**[Pa5]**-,*On Radon vector measures I*, submitted.**[Pe]**A. Pełczyński,*Projections in certain Banach spaces*, Studia Math.**19**(1960), 209–228. MR**0126145****[R]**Walter Rudin,*Functional analysis*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR**0365062****[Th]**Erik Thomas,*L’intégration par rapport à une mesure de Radon vectorielle*, Ann. Inst. Fourier (Grenoble)**20**(1970), no. 2, 55–191 (1971) (French, with English summary). MR**0463396****[Tu]**Ju. B. Tumarkin,*Locally convex spaces with basis*, Dokl. Akad. Nauk SSSR**195**(1970), 1278–1281 (Russian). MR**0271694**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
47B38,
46G10,
28B05

Retrieve articles in all journals with MSC (1991): 47B38, 46G10, 28B05

Additional Information

**T. V. Panchapagesan**

Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela

Email:
panchapa@ciens.ula.ve

DOI:
https://doi.org/10.1090/S0002-9947-98-02358-7

Keywords:
Weakly compact operators,
representing measure,
vector measure,
quasicomplete locally compact Hausdorff space,
Borel (resp. $\sigma$-Borel,
Baire) regularity,
inner regularity and outer regularity

Received by editor(s):
November 17, 1995

Additional Notes:
Supported by the C.D.C.H.T. project C-586 of the Universidad de los Andes, Mérida, and by the international cooperation project between CONICIT-Venezuela and CNR-Italy.

Dedicated:
Dedicated to Professor V. K. Balachandran on the occasion of his seventieth birthday

Article copyright:
© Copyright 1998
American Mathematical Society