On the representation of order continuous operators by random measures
HTML articles powered by AMS MathViewer
- by L. Weis PDF
- Trans. Amer. Math. Soc. 285 (1984), 535-563 Request permission
Abstract:
Using the representation $Tf(y) = \smallint f\;d{v_y}$, where $({v_y})$ is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator $T$ is (lattice-)orthogonal to all integral operators (i.e. all ${v_y}$ are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all ${v_y}$ are atomic) if and only if $T$ satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all ${v_y}$ are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.References
- William Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532. MR 365167, DOI 10.2307/1970956
- George R. Barnes and Robert Whitley, The oscillation of an operator on $L^{p}$, Trans. Amer. Math. Soc. 211 (1975), 339–351. MR 405158, DOI 10.1090/S0002-9947-1975-0405158-6
- J. Bourgain, Dunford-Pettis operators on $L^{1}$ and the Radon-Nikodým property, Israel J. Math. 37 (1980), no. 1-2, 34–47. MR 599300, DOI 10.1007/BF02762866
- James R. Brown, Approximation theorems for Markov operators, Pacific J. Math. 16 (1966), 13–23. MR 192552
- A. V. Buhvalov, The integral representation of linear operators, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 5–14, 184, 191 (Russian, with English summary). Investigations on linear operators and the theory of functions, V. MR 0399929
- Michèle Capon, Primarité de $l_{p}(L^{1})$, Math. Ann. 250 (1980), no. 1, 55–63 (French). MR 581631, DOI 10.1007/BF01422184
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- Raouf Doss, Convolution of singular measures, Studia Math. 45 (1973), 111–117. (errata insert). MR 328474, DOI 10.4064/sm-45-2-111-117
- Lester Dubins and David Freedman, Measurable sets of measures, Pacific J. Math. 14 (1964), 1211–1222. MR 174687
- 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
- P. Enflo and T. W. Starbird, Subspaces of $L^{1}$ containing $L^{1}$, Studia Math. 65 (1979), no. 2, 203–225. MR 557491, DOI 10.4064/sm-65-2-203-225
- Hicham Fakhoury, Représentations d’opérateurs à valeurs dans $L^{1}(X,\,\Sigma ,\,\mu )$, Math. Ann. 240 (1979), no. 3, 203–212 (French). MR 526843, DOI 10.1007/BF01362310
- Shaul R. Foguel, The ergodic theory of Markov processes, Van Nostrand Mathematical Studies, No. 21, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0261686
- D. H. Fremlin, A positive compact operator, Manuscripta Math. 15 (1975), no. 4, 323–327. MR 385618, DOI 10.1007/BF01486602
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606
- Cassius Ionescu Tulcea, Sur certains endomorphismes de $L_{C}{}^{\infty }\,(Z,\,\mu )$, C. R. Acad. Sci. Paris 261 (1965), 4961–4963 (French). MR 196512
- W. B. Johnson and L. Jones, Every $L_{p}$ operator is an $L_{2}$ operator, Proc. Amer. Math. Soc. 72 (1978), no. 2, 309–312. MR 507330, DOI 10.1090/S0002-9939-1978-0507330-1
- M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
- N. J. Kalton, The endomorphisms of $L_{p}(0\leq p\leq i)$, Indiana Univ. Math. J. 27 (1978), no. 3, 353–381. MR 470670, DOI 10.1512/iumj.1978.27.27027 —, Linear operators on ${L_p}$ for $0 \leqslant p \leqslant 1$, Trans. Amer. Math. Soc. 259 (1980), 317-355.
- N. J. Kalton, Embedding $L_{1}$ in a Banach lattice, Israel J. Math. 32 (1979), no. 2-3, 209–220. MR 531264, DOI 10.1007/BF02764917
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144 M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik and E. B. Sbolevskii, Integral operators in spaces of summable functions, Nordhoff, Leyden, 1976.
- S. Kwapień, On the form of a linear operator in the space of all measurable functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 951–954 (English, with Russian summary). MR 336313
- H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
- Heinrich P. Lotz, Extensions and liftings of positive linear mappings on Banach lattices, Trans. Amer. Math. Soc. 211 (1975), 85–100. MR 383141, DOI 10.1090/S0002-9947-1975-0383141-7
- W. A. J. Luxemburg and A. C. Zaanen, The linear modulus of an order bounded linear transformation. I, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 422–434. MR 0303337 —, Riesz spaces, North-Holland, Amsterdam and London, 1971. W. A. J. Luxemburg and A. R. Schep, A Radon-Nikodym type theorem for positive operators and a dual, Nederl. Akad. Wetensch. Proc. Ser. A 81 (1978).
- W. A. J. Luxemburg, Some aspects of the theory of Riesz spaces, University of Arkansas Lecture Notes in Mathematics, vol. 4, University of Arkansas, Fayetteville, Ark., 1979. MR 568706
- R. Daniel Mauldin, David Preiss, and Heinrich von Weizsäcker, Orthogonal transition kernels, Ann. Probab. 11 (1983), no. 4, 970–988. MR 714960
- Rainer J. Nagel and Ulf Schlotterbeck, Integraldarstellung regulärer Operatoren auf Banachverbänden, Math. Z. 127 (1972), 293–300 (German). MR 313868, DOI 10.1007/BF01114932 H. Neunzert, An introduction to nonlinear Boltzmann-Vlasov equation, Lecture Notes Internat. Summer School, Montecatini, 1981.
- Jacques Neveu, Bases mathématiques du calcul des probabilités, Masson et Cie, Éditeurs, Paris, 1964 (French). MR 0198504
- K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
- A. Pełczyński, On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular operators in $L(\nu )$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 37–41. MR 177301
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- A. R. Schep, Kernel operators, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 1, 39–53. MR 528217
- A. R. Sourour, Pseudo-integral operators, Trans. Amer. Math. Soc. 253 (1979), 339–363. MR 536952, DOI 10.1090/S0002-9947-1979-0536952-2
- A. R. Sourour, Characterization and order properties of pseudo-integral operators, Pacific J. Math. 99 (1982), no. 1, 145–158. MR 651492 T. Starbird, Subspaces of ${L_1}$ containing ${L_1}$, Ph.D. thesis, University of California, Berkeley, 1976.
- B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki. MR 0224522
- Lutz Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), no. 1, 83–96. MR 642619, DOI 10.1512/iumj.1982.31.31010
- L. Weis, A note on diffuse random measures, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 2, 239–244. MR 722130, DOI 10.1007/BF00532481
- Lutz W. Weis, Decompositions of positive operators and some of their applications, Functional analysis: surveys and recent results, III (Paderborn, 1983) North-Holland Math. Stud., vol. 90, North-Holland, Amsterdam, 1984, pp. 95–115. MR 761375, DOI 10.1016/S0304-0208(08)71469-5
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 285 (1984), 535-563
- MSC: Primary 47B38; Secondary 60G57
- DOI: https://doi.org/10.1090/S0002-9947-1984-0752490-8
- MathSciNet review: 752490