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Positivity pp 127-160 | Cite as

Vector Measures, Integration and Applications

  • G. P. Curbera
  • W. J. Ricker
Part of the Trends in Mathematics book series (TM)

Abstract

We will deal exclusively with the integration of scalar (i.e., ℝ or ℂ)-valued functions with respect to vector measures. The general theory can be found in [36, 37, 32], [44, Ch. I II] and [67, 124], for example. For applications beyond these texts we refer to [38, 66, 80, 102, 117] and the references therein, and the survey articles [33, 68]. Each of these references emphasizes its own preferences, as will be the case with this article. Our aim is to present some theoretical developments over the past 15 years or so (see §1) and to highlight some recent applications. Due to space limitation we restrict the applications to two topics. Namely, the extension of certain operators to their optimal domain (see §2) and aspects of spectral integration (see §3). The interaction between order and positivity with properties of the integration map of a vector measure (which is defined on a function space) will become apparent and plays a central role.

Keywords

Banach Space Boolean Algebra Spectral Measure Banach Lattice Vector Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. Albrecht and W.J. Ricker, Local spectral properties of constant coefficient differential operators in L p(ℝN) J. Operator Theory 24(1990), 85–103.MATHMathSciNetGoogle Scholar
  2. [2]
    E. Albrecht and W.J. Ricker, Local spectral properties of certain matrix differential operators in L p(ℝN)m, J. Operator Theory 35 (1996), 3–37.MATHMathSciNetGoogle Scholar
  3. [3]
    C.D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, Academic Press, New York-London, 1978.MATHGoogle Scholar
  4. [4]
    N. Aronzajn and P. Szeptycki, On general integral transformations, Math. Ann. 163 (1966), 127–154.MathSciNetGoogle Scholar
  5. [5]
    W.G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc., 80 (1955), 345–459.MATHMathSciNetGoogle Scholar
  6. [6]
    C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, 1988.MATHGoogle Scholar
  7. [7]
    E. Berkson and T.A. Gillespie, Spectral decompositions and harmonic analysis in UMD spaces, Studia Math. 112 (1994), 13–49.MATHMathSciNetGoogle Scholar
  8. [8]
    J. Bonet and S. Diaz-Madrigal, Ranges of vector measures in Fréchet spaces, Indag Math. (N.S.), 11 (2000), 19–30.MATHMathSciNetGoogle Scholar
  9. [9]
    J. Bonet and W.J. Ricker, Boolean algebras of projections in (DF)-and (LF)-spaces, Bull. Austral. Math. Soc., 67 (2003), 297–303.MATHMathSciNetGoogle Scholar
  10. [10]
    J. Bonet and W.J. Ricker, Spectral measures in classes of Fréchet spaces, Bull. Soc. Roy. Sci. Liège, 73 (2004), 99–117.MATHMathSciNetGoogle Scholar
  11. [11]
    J. Bonet and W.J. Ricker, The canonical spectral measure in Köthe echelon spaces, Integral Equations Operator Theory, 53 (2005), 477–496.MATHMathSciNetGoogle Scholar
  12. [12]
    J. Bonet, S. Okada and W.J. Ricker, The canonical spectral measure and Köthe function spaces, Quaestiones Math. 29 (2006), 91–116.MATHMathSciNetGoogle Scholar
  13. [13]
    J. Bonet and W.J. Ricker, Schauder decompositions and the Grothendieck and Dunford-Pettis properties in Köthe echelon spaces of infinite order, Positivity, 11 (2007), 77–93.MATHMathSciNetGoogle Scholar
  14. [14]
    J. Bourgain, Some remarks on Banach spaces in which martingale differences are unconditional, Ark. Math. 21 (1983), 163–168.MATHMathSciNetGoogle Scholar
  15. [15]
    A. Cianchi and L. Pick, Sobolev embeddings into BMO, VMO, and L spaces, Ark. Mat., 36 (1998), 317–340.MATHMathSciNetGoogle Scholar
  16. [16]
    G.P. Curbera, El espacio de funciones integrables respecto de una medida vectorial, Ph.D. Thesis, Univ. of Sevilla, 1992.Google Scholar
  17. [17]
    G.P. Curbera, Operators into L 1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330.MATHMathSciNetGoogle Scholar
  18. [18]
    G.P. Curbera, When L 1 of a vector measure is an AL-space, Pacific J. Math. 162 (1994), 287–303.MATHMathSciNetGoogle Scholar
  19. [19]
    G.P. Curbera, Banach space properties of L 1 of a vector measure, Proc. Amer. Math. Soc. 123 (1995), 3797–3806.MATHMathSciNetGoogle Scholar
  20. [20]
    G.P. Curbera and W.J. Ricker, Optimal domains for kernel operators via interpolation, Math. Nachr., 244 (2002), 47–63.MATHMathSciNetGoogle Scholar
  21. [21]
    G.P. Curbera and W.J. Ricker, Optimal domains for the kernel operator associated with Sobolev’s inequality, Studia Math., 158 (2003), 131–152.MATHMathSciNetGoogle Scholar
  22. [22]
    G.P. Curbera and W.J. Ricker, Corrigenda to “Optimal domains for the kernel operator associated with Sobolev’s inequality”, Studia Math., 170 (2005) 217–218.MATHMathSciNetGoogle Scholar
  23. [23]
    G.P. Curbera and W.J. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.), 17 (2006), 187–204.MATHMathSciNetGoogle Scholar
  24. [24]
    G.P. Curbera and W.J. Ricker, Compactness properties of Sobolev imbeddings for rearrangement invariant norms, Trans. Amer. Math. Soc., 359 (2007), 1471–1484.MATHMathSciNetGoogle Scholar
  25. [25]
    G.P. Curbera and W.J. Ricker, Can optimal rearrangement invariant Sobolev imbeddings be further extended?, Indiana Univ. Math. J., 56 (2007), 1489–1497.MathSciNetGoogle Scholar
  26. [26]
    G.P. Curbera and W.J. Ricker, The Fatou property in p-convex Banach lattices, J. Math. Anal. Appl., 328 (2007), 287–294.MATHMathSciNetGoogle Scholar
  27. [27]
    O. Delgado, Banach function subspaces of L 1 of a vector measure and related Orlicz spaces, Indag. Math. (N.S.), 15 (2004), 485–495.MATHMathSciNetGoogle Scholar
  28. [28]
    O. Delgado, L 1-spaces for vector measures defined on δ-rings, Archiv. Math. (Basel) 84 (2005), 43–443.MathSciNetGoogle Scholar
  29. [29]
    O. Delgado, Optimal domains for kernel operators on [0,∞)×[0,∞), Studia Math. 174 (2006), 131–145.MATHMathSciNetGoogle Scholar
  30. [30]
    O. Delgado and J. Soria, Optimal domains for the Hardy operator, J. Funct. Anal. 244 (2007), 119–133.MATHMathSciNetGoogle Scholar
  31. [31]
    J.C. Diaz, A. Fernández and F. Naranjo, Fréchet AL-spaces have the Dunford-Pettis property, Bull. Austral. Math. Soc., 58 (1998), 383–386.MATHMathSciNetGoogle Scholar
  32. [32]
    J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.MATHGoogle Scholar
  33. [33]
    J. Diestel and J.J. Uhl, Jr., Progress in vector measures: 1977–83, Lecture Notes Math. 1033, Springer, Berlin Heidelberg, 1984, pp. 144–192.Google Scholar
  34. [34]
    J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics 43, Cambridge University Press, 1995.Google Scholar
  35. [35]
    J. Diestel and W.J. Ricker, The strong closure of Boolean algebras of projections in Banach spaces, J. Austral. Math. Soc. 77 (2004), 365–369.MATHMathSciNetGoogle Scholar
  36. [36]
    N. Dinculeanu, Vector Measures, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966.MATHGoogle Scholar
  37. [37]
    N. Dinculeanu, Integration on Locally Compact Spaces, Noordhoff, Leyden, 1974.Google Scholar
  38. [38]
    N. Dinculeanu, Vector Integration and Stochastic Integration in Banach spaces, Wiley-Interscience, New York, 2000.MATHGoogle Scholar
  39. [39]
    P.G. Dodds and B. dePagter, Orthomorphisms and Boolean algebras of projections, Math. Z. 187 (1984), 361–381.MATHMathSciNetGoogle Scholar
  40. [40]
    P.G. Dodds and W.J. Ricker, Spectral measures and the Bade reflexivity theorem, J. Funct. Anal., 61 (1985), 136–163.MATHMathSciNetGoogle Scholar
  41. [41]
    P.G. Dodds, B. dePagter and W.J. Ricker, Reflexivity and order properties of scalar-type spectral operators in locally convex spaces, Trans. Amer. Math. Soc. 293 (1986), 355–380.MATHMathSciNetGoogle Scholar
  42. [42]
    H.R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978.MATHGoogle Scholar
  43. [43]
    H.R. Dowson, M.B. Ghaemi and P.G. Spain, Boolean algebras of projections and algebras of spectral operators, Pacific J. Math. 209 (2003), 1–16.MATHMathSciNetGoogle Scholar
  44. [44]
    N. Dunford and J.T. Schwartz, Linear Operators I: General Theory (2nd Ed), Wiley-Interscience, New York, 1964.Google Scholar
  45. [45]
    N. Dunford and J.T. Schwartz, Linear Operators III: Spectral Operators, Wiley-Interscience, New York, 1971.MATHGoogle Scholar
  46. [46]
    D. van Dulst, Characterizations of Banach Spaces not Containing1, CWI Tract 59, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.Google Scholar
  47. [47]
    D. Edmunds, R. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307–355.MATHMathSciNetGoogle Scholar
  48. [48]
    A. Fernández and F. Naranjo, Rybakov’s theorem for vector measures in Fréchet spaces, Indag. Math. (N.S.), 8 (1997), 33–42.MATHMathSciNetGoogle Scholar
  49. [49]
    A. Fernández and F. Naranjo, Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure, J. Austral. Math. Soc. (Ser. A), 65 (1998), 176–193.MATHGoogle Scholar
  50. [50]
    A. Fernández, F. Naranjo and W.J. Ricker, Completeness of L 1-spaces for measures with values in complex vector spaces, J. Math. Anal. Appl. 223 (1998), 76–87.MATHMathSciNetGoogle Scholar
  51. [51]
    A. Fernández, F. Mayoral, F. Naranjo and P.J. Paul, Weakly sequentially complete Fréchet spaces of integrable functions, Arch. Math. (Basel), 71 (1998), 223–228.MATHGoogle Scholar
  52. [52]
    A. Fernández and F. Naranjo, Strictly positive linear functionals and representation of Fréchet lattices with the Lebesgue property, Indag. Math. (N.S.), 10 (1999), 381–391.Google Scholar
  53. [53]
    A. Fernández and F. Naranjo, AL-and AM-spaces of integrable scalar functions with respect to a Fréchet-valued measure, Quaestiones Math. 23 (2000), 247–258.MATHGoogle Scholar
  54. [54]
    A. Fernández and F. Naranjo, Nuclear Fréchet lattices, J. Austral. Math. Soc. 72 (2002), 409–417.MATHGoogle Scholar
  55. [55]
    A. Fernández, F. Mayoral, F. Naranjo, C. Sáez and E.A. Sánchez-Pérez, Vector measure Maurey-Rosenthal-type factorizations and ℓ-sums of L 1-spaces, J. Funct. Anal. 220 (2005), 460–485.MATHMathSciNetGoogle Scholar
  56. [56]
    A. Fernández, F. Mayoral, F. Naranjo, C. Sáez and E.A. Sánchez-Pérez, Spaces of p-integrable functions with respect to a vector measure, Positivity, 10 (2006), 1–16.MATHMathSciNetGoogle Scholar
  57. [57]
    U. Fixman, Problems in spectral operators, Pacific J. Math. 9 (1959), 1029–1051.MATHMathSciNetGoogle Scholar
  58. [58]
    D.H. Fremlin, B. dePagter and W.J. Ricker, Sequential closedness of Boolean algebras of projections in Banach spaces, Studia Math. 167 (2005), 45–62.MATHMathSciNetGoogle Scholar
  59. [59]
    D.H. Fremlin and D. Preiss, On a question of W.J. Ricker, Electronic file (December, 2006); http://www.essex.ac.uk/maths/staff/fremlin/n05403.ps.
  60. [60]
    G.I. Gaudry and W.J. Ricker, Spectral properties of L p translations, J. Operator Theory, 14 (1985), 87–111.MATHMathSciNetGoogle Scholar
  61. [61]
    G.I. Gaudry and W.J. Ricker, Spectral properties of translation operators in certain function spaces, Illinois J. Math., 31 (1987), 453–468.MATHMathSciNetGoogle Scholar
  62. [62]
    G.I. Gaudry, B.R.F. Jefferies and W.J. Ricker, Vector-valued multipliers: convolution with operator-valued measures, Dissertationes Math., 385 (2000), 1–77.MathSciNetGoogle Scholar
  63. [63]
    T.A. Gillespie, A spectral theorem for L p translations, J. London Math. Soc. (2)11 (1975), 499–508.MATHMathSciNetGoogle Scholar
  64. [64]
    T.A. Gillespie, Strongly closed bounded Boolean algebras of projections, Glasgow Math. J., 22 (1981), 73–75.MATHMathSciNetGoogle Scholar
  65. [65]
    T.A. Gillespie, Boundedness criteria for Boolean algebras of projections, J. Funct. Anal. 148 (1997), 70–85.MATHMathSciNetGoogle Scholar
  66. [66]
    B. Jefferies, Evolution Process and the Feynman-Kac Formula, Kluwer, Dordrecht, 1996.Google Scholar
  67. [67]
    I. Kluvánek and G. Knowles, Vector Measures and Control Systems, North-Holland, Amsterdam, 1976.MATHGoogle Scholar
  68. [68]
    I. Kluvánek, Applications of vector measures, In: Integration, topology, and geometry in linear spaces, Proc. Conf. Chapel Hill/N.C. 1979, Contemp. Math. 2 (1980), 101–134.Google Scholar
  69. [69]
    G.L. Krabbe, Convolution operators which are not of scalar type, Math. Z. 69 (1958), 346–350.MATHMathSciNetGoogle Scholar
  70. [70]
    S.G. Krein, Ju. I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence, 1982.Google Scholar
  71. [71]
    I. Labuda and P. Szeptycki, Extended domains of some integral operators with rapidly oscillating kernels, Indag. Math. 48 (1986), 87–98.MathSciNetGoogle Scholar
  72. [72]
    R. Larsen, An Introduction to the Theory of Multipliers, Springer-Verlag, Berlin Heidelberg New York, 1971.MATHGoogle Scholar
  73. [73]
    D.R. Lewis, On integrability and summability in vector spaces, Illinois J. Math., 16 (1972), 294–307.MATHMathSciNetGoogle Scholar
  74. [74]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces vol. II, Springer-Verlag, Berlin, (1979).MATHGoogle Scholar
  75. [75]
    W.A. Luxemburg and A.C. Zaanen, Notes on Banach function spaces, I, Nederl. Akad. Wet., Proc., 66 =Indag. Math. 25 (1963) 135–147.MathSciNetGoogle Scholar
  76. [76]
    W.A. Luxemburg and A.C. Zaanen, Notes on Banach function spaces, II, Nederl. Akad. Wet., Proc., 66 = Indag. Math. 25 (1963) 148–153.MathSciNetGoogle Scholar
  77. [77]
    W.A. Luxemburg and A.C. Zaanen, Notes on Banach function spaces, III, Nederl. Akad. Wet., Proc., 66 = Indag. Math. 25 (1963) 239–250.MathSciNetGoogle Scholar
  78. [78]
    W.A. Luxemburg and A.C. Zaanen, Notes on Banach function spaces, IV, Nederl. Akad. Wet., Proc., 66 = Indag. Math. 25 (1963) 251–263.MathSciNetGoogle Scholar
  79. [79]
    W.A. Luxemburg, Spaces of measurable functions, Jeffery-Williams Lectures 1968–72 Canad. Math. Congr. (1972) 45–71.Google Scholar
  80. [80]
    T.-W. Ma, Banach Hilbert Spaces, Vector Measures and Group Representations, World Scientific, Singapore, 2002.MATHGoogle Scholar
  81. [81]
    P. R. Masani and H. Niemi, The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on δ-rings, Adv. Math. 73 (1989), 204–241.MATHMathSciNetGoogle Scholar
  82. [82]
    P.R. Masani and H. Niemi, The integration theory of Banach space-valued measures and the Tonelli-Fubini theorems. II. Pettis integration, Adv. Math. 75 (1989), 121–167.MathSciNetGoogle Scholar
  83. [83]
    C.A. McCarthy, Commuting Boolean algebras of projections, Pacific J. Math. 11 (1961), 295–307.MATHMathSciNetGoogle Scholar
  84. [84]
    G. Mockenhaupt and W.J. Ricker, Idempotent multipliers for L p (ℜ), Arch. Math. (Basel), 74 (2000), 61–65.MATHMathSciNetGoogle Scholar
  85. [85]
    G. Mockenhaupt and W.J. Ricker, Fuglede’s theorem, the bicommutant theorem and p-multiplier operators for the circle, J. Operator Theory, 49 (2003), 295–310.MATHMathSciNetGoogle Scholar
  86. [86]
    G. Mockenhaupt and W.J. Ricker, Approximation of p-multiplier operators via their spectral projections, Positivity (to appear).Google Scholar
  87. [87]
    G. Mockenhaupt and W.J. Ricker, Optimal extension of the Hausdorff-Young inequality, J. Reine Angew. Math. (to appear).Google Scholar
  88. [88]
    G. Muraz and P. Szeptycki, Domains of trigonometric transforms, Rocky Mountain J. Math. 26 (1996), 1517–1527.MATHMathSciNetGoogle Scholar
  89. [89]
    K.K. Oberai, Sum and product of commuting spectral operators, Pacific J. Math., 25 (1968), 129–146.MATHMathSciNetGoogle Scholar
  90. [90]
    S. Okada, Spectrum of scalar-type spectral operators and Schauder decompositions, Math. Nachr. 139 (1988), 167–174.MATHMathSciNetGoogle Scholar
  91. [91]
    S. Okada and W.J. Ricker, Vector measures and integration in non-complete spaces, Arch. Math. (Basel), 63 (1994), 344–353.MATHMathSciNetGoogle Scholar
  92. [92]
    S. Okada and W.J. Ricker, Boolean algebras of projections and ranges of spectral measures. Dissertationes Math., 365, 33p., 1997.Google Scholar
  93. [93]
    S. Okada and W.J. Ricker, Representation of complete Boolean algebras of projections as ranges of spectral measures, Acta Sci. Math. (Szeged), 63 (1997), 209–227 and 63 (1997), 689–693.MATHMathSciNetGoogle Scholar
  94. [94]
    S. Okada and W.J. Ricker, Criteria for closedness of spectral measures and completeness of Boolean algebras of projections, J. Math. Anal. Appl., 232 (1999), 197–221.MATHMathSciNetGoogle Scholar
  95. [95]
    S. Okada and W.J. Ricker, Integration with respect to the canonical spectral measure in sequence spaces, Collect. Math. 50 (1999), 95–118.MATHMathSciNetGoogle Scholar
  96. [96]
    S. Okada, W.J. Ricker and L. Rodríguez-Piazza, Compactness of the integration operator associated with a vector measure, Studia Math., 150 (2002), 133–149.MATHMathSciNetGoogle Scholar
  97. [97]
    S. Okada and W.J. Ricker, Compact integration operators for Fréchet-space-valued measures, Indag. Math. (New Series), 13 (2002), 209–227.MATHMathSciNetGoogle Scholar
  98. [98]
    S. Okada and W.J. Ricker, Fréchet-space-valued measures and the AL-property, Rev. R. Acad. Cien. Serie A. Mat. RACSAM, 97 (2003), 305–314.MATHMathSciNetGoogle Scholar
  99. [99]
    S. Okada and W.J. Ricker, Optimal domains and integral representations of convolution operators in L p(G), Integral Equations Operator Theory, 48 (2004), 525–546.MATHMathSciNetGoogle Scholar
  100. [100]
    S. Okada and W.J. Ricker, Optimal domains and integral representations of L p(G)-valued convolution operators via measures, Math. Nachr. 280 (2007), 423–436.MATHMathSciNetGoogle Scholar
  101. [101]
    S. Okada, W.J. Ricker and E.A. Sánchez-Pérez, Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory Advances and Applications, Birkhäuser Verlag, 2008 (to appear).Google Scholar
  102. [102]
    E. Pap (Ed.), Handbook of Measure Theory I, Part 2: Vector Measures, North-Holland, Amsterdam, 2002, pp. 345–502.Google Scholar
  103. [103]
    B. dePagter and W.J. Ricker, Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators, Proc. Edinburgh Math. Soc. 40 (1997) 425–435.MathSciNetGoogle Scholar
  104. [104]
    B. dePagter and W.J. Ricker, Products of commuting Boolean algebras of projections and Banach space geometry, Proc. London Math. Soc. (3)91 (2005), 483–508.MathSciNetGoogle Scholar
  105. [105]
    B. dePagter and W.J. Ricker, R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry, In: Proceedings of the Conference “Positivity IV-Theory and Applications”, July 2005, Eds. M. Weber and J. Voigt, Technische Universität Dresden, Germany, pp. 115–129 (2006).Google Scholar
  106. [106]
    B. dePagter and W.J. Ricker, C(K)-representations and R-boundedness, J. London Math. Soc. (to appear).Google Scholar
  107. [107]
    B. dePagter and W.J. Ricker, R-bounded representations of L 1 (G), Positivity (to appear).Google Scholar
  108. [108]
    L. Pick, Optimal Sobolev Embeddings, Rudolph-Lipschitz-Vorlesungsreihe Nr. 43, SFB 256: Nichtlineare Partielle Differentialgleichungen, Univ. of Bonn, 2002.Google Scholar
  109. [109]
    G. Pisier, Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), 3–19.MATHMathSciNetGoogle Scholar
  110. [110]
    W.J. Ricker, Spectral operators of scalar-type in Grothendieck spaces with the Dunford-Pettis property, Bull. London Math. Soc. 17 (1985), 268–270.MATHMathSciNetGoogle Scholar
  111. [111]
    W.J. Ricker, Spectral like multipliers in Lp(ℝ), Arch. Math. (Basel), 57 (1991), 395–401.MATHMathSciNetGoogle Scholar
  112. [112]
    W.J. Ricker, Well bounded operators of type (B) in H.I. spaces, Acta Sci. Math. (Szeged) 59 (1994), 475–488.MATHMathSciNetGoogle Scholar
  113. [113]
    W.J. Ricker, Spectrality for matrices of Fourier multiplier operators acting in Lp-spaces over lca groups, Quaestiones Math. 19 (1996), 237–257.MATHMathSciNetGoogle Scholar
  114. [114]
    W.J. Ricker, Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces, Proc. Amer. Math. Soc. 125 (1997), 2401–2407.MATHMathSciNetGoogle Scholar
  115. [115]
    W.J. Ricker, The sequential closedness of σ-complete Boolean algebras of projections, J. Math. Anal. Appl., 208 (1997), 364–371.MATHMathSciNetGoogle Scholar
  116. [116]
    W.J. Ricker, The strong closure of σ-complete Boolean algebras of projections, Arch. Math. (Basel), 72 (1999), 282–288.MATHMathSciNetGoogle Scholar
  117. [117]
    W.J. Ricker, Operator Algebras Generated by Commuting Projections: A Vector Measure Approach, Lecture Notes Math. 1711, Springer, Berlin Heidelberg, 1999.MATHGoogle Scholar
  118. [118]
    W.J. Ricker, Resolutions of the identity in Fréchet spaces, Integral Equations Operator Theory, 41 (2001), 63–73.MATHMathSciNetGoogle Scholar
  119. [119]
    W.J. Ricker and M. Väth, Spaces of complex functions and vector measures in incomplete spaces, J. Function Spaces Appl., 2 (2004), 1–16.MATHGoogle Scholar
  120. [120]
    L. Rodríguez-Piazza, Derivability, variation and range of a vector measure, Studia Math., 112 (1995), 165–187.MATHMathSciNetGoogle Scholar
  121. [121]
    L. Rodríguez-Piazza and C. Romero-Moreno, Conical measures and properties of a vector measure determined by its range, Studia Math., 125 (1997), 255–270.MathSciNetGoogle Scholar
  122. [122]
    E.A. Sánchez-Pérez, Compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces, Illinois J. Math., 45 (2001), 907–923.MATHMathSciNetGoogle Scholar
  123. [123]
    H.H. Schaefer and B. Walsh, Spectral operators in spaces of distributions, Bull. Amer. Math. Soc., 68 (1962), 509–511.MATHMathSciNetGoogle Scholar
  124. [124]
    K.D. Schmidt, Jordan Decompositions of Generalized Vector Measures, Longman Scientific & Technical, Harlow, 1989.MATHGoogle Scholar
  125. [125]
    M.A. Sofi, Vector measures and nuclear operators, Illinois J. Math. 49 (2005), 369–383.MATHMathSciNetGoogle Scholar
  126. [126]
    M.A. Sofi, Absolutely p-summable sequences in Banach spaces and range of vector measures, Rocky Mountain J. Math., to appear.Google Scholar
  127. [127]
    M.A. Sofi, Fréchet-valued measures and nuclearity, Houston J. Math., to appear.Google Scholar
  128. [128]
    G. Stefansson, L 1 of a vector measure, Le Mathematiche 48 (1993), 219–234.MATHMathSciNetGoogle Scholar
  129. [129]
    P. Szeptycki, Notes on integral transformations, Dissertationes Math. 231 (1984), 48pp.Google Scholar
  130. [130]
    P. Szeptycki, Extended domains of some integral operators, Rocky Mountain J. Math. 22 (1992), 393–404.MATHMathSciNetGoogle Scholar
  131. [131]
    U.B. Tewari, Vector-valued multipliers, J. Anal. 12 (2004), 99–105.MATHMathSciNetGoogle Scholar
  132. [132]
    E. Thomas, The Lebesgue-Nikodým theorem for vector valued Radon measures, Mem. Amer. Math. Soc. 139 (1974).Google Scholar
  133. [133]
    A.I. Veksler, Cyclic Banach spaces and Banach lattices, Soviet Math. Dokl. 14 (1973), 1773–1779.Google Scholar
  134. [134]
    B. Walsh, Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 (1965), 295–326.MATHMathSciNetGoogle Scholar
  135. [135]
    B. Walsh, Spectral decomposition of quasi-Montel sapces, Proc. Amer. Math. Soc., (2) 17 (1966), 1267–1271.MATHMathSciNetGoogle Scholar
  136. [136]
    A. C. Zaanen, Integration, 2nd rev. ed. North-Holland, Amsterdam; Interscience, New York Berlin, 1967.MATHGoogle Scholar
  137. [137]
    A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.MATHGoogle Scholar

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© Birkhäuser Verlag AG2007 2007

Authors and Affiliations

  • G. P. Curbera
    • 1
  • W. J. Ricker
    • 2
  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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