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Transactions of the American Mathematical Society

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Linear operators and vector measures


Authors: J. K. Brooks and P. W. Lewis
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
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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.


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  • [1] R. G. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337-352. MR 18, 289. MR 0080721 (18:289a)
  • [2] R. G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289-305. MR 16, 1123. MR 0070050 (16:1123c)
  • [3] J. Batt and E. J. Berg, Linear bounded transformations on the space of continuous functions, J. Functional Analysis 4 (1969), 215-239. MR 40 #1798. MR 0248546 (40:1798)
  • [4] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 22 #5872. MR 0115069 (22:5872)
  • [5] J. K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999-1001. MR 44 #4178. MR 0286971 (44:4178)
  • [6] -, Weak compactness in the space of vector measures, Bull. Amer. Math. Soc. 78 (1972), 284-287. MR 0324408 (48:2760)
  • [7] -, Contributions to the theory of finitely additive measures, Advances in Math. (to appear).
  • [8] -, Equicontinuous sets of measures and applications to Vitali's integral convergence theorem and control measures, Advances in Math. 10 (1973), 165-171. MR 0320268 (47:8807)
  • [9] J. K. Brooks and R. S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294-1298. MR 42 #4697. MR 0269802 (42:4697)
  • [10] J. K. Brooks and P. W. Lewis, Operators on function spaces, Bull. Amer. Math. Soc. 78 (1972), 697-701. MR 0298442 (45:7494)
  • [11] N. Dinculeanu, Vector measures, Internat. Series of Monographs in Pure and Appl. Math., vol. 95, Pergamon Press, New York; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 34 #6011b. MR 0206190 (34:6011b)
  • [12] N. Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323-392. MR 1, 338. MR 0002020 (1:338b)
  • [13] N. Dunford and J. T. Schwartz, Linear operators. I. General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [14] J. R. Edwards and S. G. Wayment, A unifying representation theorem, Math. Ann. 187 (1970), 317-328. MR 42 #5074. MR 0270181 (42:5074)
  • [15] C. Foiaş and I. 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 24 #A1618. MR 0131770 (24:A1618)
  • [16] R. K. Goodrich, A Riesz representation theorem, Proc. Amer. Math. Soc. 24 (1970), 629-636. MR 0415386 (54:3474)
  • [17] A. Grothendieck, Sur les applications linéaires faiblement compactes d'espaces du type $ C(K)$, Canad. J. Math. 5 (1953), 129-173. MR 15, 438. MR 0058866 (15:438b)
  • [18] L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [19] P. R. Halmos, Measure theory, Van Nostrand, Princeton, N.J., 1950. MR 11, 504. MR 0033869 (11:504d)
  • [20] L. Hormander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221. MR 0248435 (40:1687)
  • [21] P. W. Lewis, Extension of operator valued set junctions with finite semivariation, Proc. Amer. Math. Soc. 22 (1969), 563-569. MR 39 #7061. MR 0245755 (39:7061)
  • [22] -, Some regularity conditions on vector measures with finite semivariation, Rev. Roumaine Math. Pures Appl. 15 (1970), 375-384. MR 41 #8626. MR 0264027 (41:8626)
  • [23] -, Vector measures and topology, Rev. Roumaine Math. Pures Appl. 16 (1971), 1201-1209. MR 0308358 (46:7472)
  • [24] -, Addendum to: Vector measures and topology, Rev. Roumaine Math. Pures Appl. 16 (1971), 1211-1213. MR 0308358 (46:7472)
  • [25] -, Regularity conditions and absolute continuity for vector measures, J. Reine Angew. Math. 247 (1971), 80-86. MR 43 #4997. MR 0279274 (43:4997)
  • [26] -, Variational semiregularity and norm convergence, J. Reine Angew. Math. (to appear).
  • [27] R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943), 108-136. MR 4, 218. MR 0007938 (4:218f)
  • [28] C. E. Rickart, Decompositions of additive set functions, Duke Math. J. 10 (1943), 653-665. MR 5, 232. MR 0009977 (5:232c)
  • [29] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 35 #1420. MR 0210528 (35:1420)
  • [30] H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [31] I. Singer, Sur les applications linéaires intégrales des espaces de fonctions continues. I, Rev. Math. Pures Appl. 4 (1959), 391-401. MR 22 #5883. MR 0115080 (22:5883)
  • [32] K. Swong, A representation theory of continuons linear maps, Math. Ann. 155 (1964), 270-291; errata, ibid. 157 (1964), 178. MR 29 #2642. MR 0165358 (29:2642)
  • [33] E. O. Thorp and R. J. Whitley, Operator representation theorems, Illinois J. Math. 9 (1965), 595-601. MR 31 #6126. MR 0181900 (31:6126)
  • [34] D. 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 33 #7865. MR 0199722 (33:7865)

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DOI: https://doi.org/10.1090/S0002-9947-1974-0338821-5
Article copyright: © Copyright 1974 American Mathematical Society

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