Completely continuous multipliers from L 1 ( G ) into L ∞ ( G )

G. Crombez; Willy Govaerts

Annales de l'institut Fourier (1984)

  • Volume: 34, Issue: 2, page 137-154
  • ISSN: 0373-0956

Abstract

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For a locally compact Hausdorff group G we investigate what functions in L ∞ ( G ) give rise to completely continuous multipliers T g from L 1 ( G ) into L ∞ ( G ) . In the case of a metrizable group we obtain a complete description of such functions. In particular, for G compact all g in L ∞ ( G ) induce completely continuous T g .

How to cite

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Crombez, G., and Govaerts, Willy. "Completely continuous multipliers from $L_1(G)$ into $L_\infty (G)$." Annales de l'institut Fourier 34.2 (1984): 137-154. <http://eudml.org/doc/74626>.

@article{Crombez1984,
abstract = {For a locally compact Hausdorff group $G$ we investigate what functions in $L_\infty (G)$ give rise to completely continuous multipliers $T_g$ from $L_1(G)$ into $L_\infty (G)$. In the case of a metrizable group we obtain a complete description of such functions. In particular, for $G$ compact all $g$ in $L_\infty (G)$ induce completely continuous $T_g$.},
author = {Crombez, G., Govaerts, Willy},
journal = {Annales de l'institut Fourier},
keywords = {completely continuous operator; L1(G); L-infinity (G); uniformly measurable functions; multiplier},
language = {eng},
number = {2},
pages = {137-154},
publisher = {Association des Annales de l'Institut Fourier},
title = {Completely continuous multipliers from $L_1(G)$ into $L_\infty (G)$},
url = {http://eudml.org/doc/74626},
volume = {34},
year = {1984},
}

TY - JOUR
AU - Crombez, G.
AU - Govaerts, Willy
TI - Completely continuous multipliers from $L_1(G)$ into $L_\infty (G)$
JO - Annales de l'institut Fourier
PY - 1984
PB - Association des Annales de l'Institut Fourier
VL - 34
IS - 2
SP - 137
EP - 154
AB - For a locally compact Hausdorff group $G$ we investigate what functions in $L_\infty (G)$ give rise to completely continuous multipliers $T_g$ from $L_1(G)$ into $L_\infty (G)$. In the case of a metrizable group we obtain a complete description of such functions. In particular, for $G$ compact all $g$ in $L_\infty (G)$ induce completely continuous $T_g$.
LA - eng
KW - completely continuous operator; L1(G); L-infinity (G); uniformly measurable functions; multiplier
UR - http://eudml.org/doc/74626
ER -

References

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  1. [1] G. CROMBEZ and W. GOVAERTS, Weakly compact convolution operators in L1(G), Simon Stevin, 52 (1978), 65-72. Zbl0379.43004MR80a:43008
  2. [2] G. CROMBEZ and W. GOVAERTS, Towards a classification of convolution-type operators from l1 to l∞, Canad. Math. Bull., 23 (1980), 413-419. Zbl0446.47017MR82f:47038
  3. [3] J. DIESTEL and J. J. UHL, Vector measures, Math. Surveys n° 15, Amer. Math. Soc., Providence, R.I., 1977. Zbl0369.46039MR56 #12216
  4. [4] N. DUNFORD and J. T. SCHWARTZ, Linear operators, part I, New-York, Interscience, 1958. Zbl0084.10402MR22 #8302
  5. [5] R. E. EDWARDS, Functional analysis, New-York, Holt, Rinehart and Winston, 1965. Zbl0182.16101MR36 #4308
  6. [6] R. HERMAN, Generalizations of weakly compact operators, Trans. Amer. Math. Soc., 132 (1968), 377-386. Zbl0159.43004MR36 #6976
  7. [7] E. HEWITT and K. A. ROSS, Abstract harmonic analysis, I, Berlin, Springer, 1963. Zbl0115.10603
  8. [8] A. PELCZYNSKI, On strictly singular and strictly cosingular operators, II, Bull. Acad. Polon. Sci., Sér. Sc. Math. Astronom. Phys., 13 (1965), 37-41. Zbl0138.38604MR31 #1564
  9. [9] A. PIETSCH, Operator ideals, Amsterdam, North-Holland Publ. Comp., 1980. Zbl0434.47030MR81j:47001
  10. [10] K. YLINEN, Characterizations of B(G) and B(G)∩AP(G) for locally compact groups, Proc. Amer. Math. Soc., 58 (1976), 151-157. Zbl0333.43004MR54 #13472

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