Linear operators and vector measures
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- by J. K. Brooks and P. W. Lewis
- Trans. Amer. Math. Soc. 192 (1974), 139-162
- DOI: https://doi.org/10.1090/S0002-9947-1974-0338821-5
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Abstract:
Compact and weakly compact operators on function spaces are studied. Those operators are characterized by properties of finitely additive set functions whose existence is guaranteed by Riesz representation theorems.References
- R. G. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337–352. MR 80721, DOI 10.4064/sm-15-3-337-352
- R. G. Bartle, N. Dunford, and J. Schwartz, Weak compactness and vector measures, Canadian J. Math. 7 (1955), 289–305. MR 70050, DOI 10.4153/CJM-1955-032-1
- Jürgen Batt and E. Jeffrey Berg, Linear bounded transformations on the space of continuous functions. , J. Functional Analysis 4 (1969), 215–239. MR 0248546, DOI 10.1016/0022-1236(69)90012-3
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
- James K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001. MR 286971, DOI 10.1090/S0002-9904-1971-12834-3
- James K. Brooks, Weak compactness in the space of vector measures, Bull. Amer. Math. Soc. 78 (1972), 284–287. MR 324408, DOI 10.1090/S0002-9904-1972-12960-4 —, Contributions to the theory of finitely additive measures, Advances in Math. (to appear).
- James K. Brooks, Equicontinuous sets of measures and applications to Vitali’s integral convergence theorem and control measures, Advances in Math. 10 (1973), 165–171. MR 320268, DOI 10.1016/0001-8708(73)90104-7
- James K. Brooks and Robert S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 269802, DOI 10.1073/pnas.67.3.1294
- James K. Brooks and Paul W. Lewis, Operators on function spaces, Bull. Amer. Math. Soc. 78 (1972), 697–701. MR 298442, DOI 10.1090/S0002-9904-1972-12988-4
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
- Nelson Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323–392. MR 2020, DOI 10.1090/S0002-9947-1940-0002020-4
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- J. R. Edwards and S. G. Wayment, A unifying representation theorem, Math. Ann. 187 (1970), 317–328. MR 270181, DOI 10.1007/BF01396462
- Ciprian Foiaş and Ivan Singer, Some remarks on the representation of linear operators in spaces of vector-valued continuous functions, Rev. Math. Pures Appl. 5 (1960), 729–752. MR 131770
- Robert Kent Goodrich, A Riesz representation theorem, Proc. Amer. Math. Soc. 24 (1970), 629–636. MR 415386, DOI 10.1090/S0002-9939-1970-0415386-2
- A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York, Inc., New York, 1969. Third revised printing. MR 0248435, DOI 10.1007/978-3-662-30722-9
- P. W. Lewis, Extension of operator valued set functions with finite semivariation, Proc. Amer. Math. Soc. 22 (1969), 563–569. MR 245755, DOI 10.1090/S0002-9939-1969-0245755-7
- P. W. Lewis, Some regularity conditions on vector measures with finite semi-variation, Rev. Roumaine Math. Pures Appl. 15 (1970), 375–384. MR 264027
- Paul W. Lewis, Vector measures and topology, Rev. Roumaine Math. Pures Appl. 16 (1971), 1201–1209; addendum, ibid. 16 (1971), 1211–1213. MR 308358
- Paul W. Lewis, Vector measures and topology, Rev. Roumaine Math. Pures Appl. 16 (1971), 1201–1209; addendum, ibid. 16 (1971), 1211–1213. MR 308358
- P. W. Lewis, Regularity conditions and absolute continuity for vector measures, J. Reine Angew. Math. 247 (1971), 80–86. MR 279274, DOI 10.1515/crll.1971.247.80 —, Variational semiregularity and norm convergence, J. Reine Angew. Math. (to appear).
- R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943), 108–136. MR 7938, DOI 10.2307/2371776
- C. E. Rickart, Decomposition of additive set functions, Duke Math. J. 10 (1943), 653–665. MR 9977, DOI 10.1215/S0012-7094-43-01061-0
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210528
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- Ivan Singer, Sur les applications linéaires intégrales des espaces de fonctions continues. I, Rev. Math. Pures Appl. 4 (1959), 391–401 (French). MR 115080
- Khyson Swong, A representation theory of continuous linear maps, Math. Ann. 155 (1964), 270–291; errata: 157 (1964), 178. MR 0165358, DOI 10.1007/BF01354862
- Edward O. Thorp and Robert J. Whitley, Operator representation theorems, Illinois J. Math. 9 (1965), 595–601. MR 181900
- Don H. Tucker, A representation theorem for a continuous linear transformation on a space of continuous functions, Proc. Amer. Math. Soc. 16 (1965), 946–953. MR 199722, DOI 10.1090/S0002-9939-1965-0199722-9
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 192 (1974), 139-162
- MSC: Primary 47B37; Secondary 46E40
- DOI: https://doi.org/10.1090/S0002-9947-1974-0338821-5
- MathSciNet review: 0338821