Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T13:34:48.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear operators which commute with translations. Part I: Representation theorems

Published online by Cambridge University Press:  09 April 2009

B. Brainerd  and
R. E. Edwards
B. Brainerd
Affiliation:
Department of MathematicsUniversity of Toronto
R. E. Edwards
Affiliation:
Department of Mathematics Institute of Advanced Studies Australian National University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In Part I of this paper we shall be concerned with the representation as convolutions of continuous linear operators which act on various function- spaces linked with a locally compact group and which commute with left — or right — translations; cf. the results in [12]. For completeness some known results are included whenever they follow from the general procedure. We have tried to follow simple general approaches as much as possible.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 3 , August 1966 , pp. 289 - 327
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Boehme, T. K., ‘Continuity and perfect operators’. J. London Math. Soc. 39 (1964), 355–358.CrossRefGoogle Scholar
[2]Bourbaki, N., Éléments de Mathématique. Intégration. Chapitre 5. Act. Sci. et Ind. No. 1244. Hermann, Paris (1956).Google Scholar
[3]de Leeuw, K., ‘Linear spaces with a compact group of operators’. Illinois J. Math. 2 (1958), 367–377.Google Scholar
[4]de Leeuw, K., ‘On LD multipliers’. Ann. of Math. (2) 81 (1965), 364–379.CrossRefGoogle Scholar
[5]Dieudonné, J., ‘Sur le produit de composition (II)’. J. de Math. Pure et App. 29 (3) (1960), 275–292.Google Scholar
[6]Edwards, D. A., ‘On translates of L∞ functions’. J. London Math. Soc. 36 (1961), 431–432.CrossRefGoogle Scholar
[7]Edwards, R. E., ‘On factor functions’. Pacific J. Math. 5 (1955), 367–378.CrossRefGoogle Scholar
[8]Edwards, R. E., ‘The form of the solution of the Cauchy problem over a group’. Studia Math. XIX (1960), 193–206.CrossRefGoogle Scholar
[9]Edwards, R. E., ‘Representation theorems for certain functional operators’. Pacific J. Math. 7 (1957), 1333–1339.Google Scholar
[10]Edwards, R. E., ‘Bipositive and isometric isomorphisms of certain convolution algebras’, Canad. J. Math. 17 (1965), 839–846.Google Scholar
[11]Edwards, R. E., ‘Changing signs of Fourier coefficients’. Pacific J. Math. 15 (1965), 463–475.CrossRefGoogle Scholar
[12]Edwards, R. E., ‘Convolution as bilinear and linear operators’. Canad. J. Math. 16 (1964), 275–285.CrossRefGoogle Scholar
[13]Edwards, R. E., ‘Operators commuting with translations’. Pacific J. Math. 16 (1966), 259–265.CrossRefGoogle Scholar
[14]Edwards, R. E., ‘Translates of L∞ functions and of bounded measures’. J. Australian Math. Soc. 4 (1964), 403–409.Google Scholar
[15]Edwards, R. E., ‘Supports and singular supports of pseudomeasures’. J. Australian Math. Soc. 6 (1966), 65–75.Google Scholar
[16]Edwards, R. E., Functional Analysis: Theory and Applications. Holt, Rinehart and Winston, Inc., New York (1965).Google Scholar
[17]Figà-Talamanca, A., ‘Multipliers of p-integrable functions’. Bull. Amer. Math. Soc. (5) 70 (1964), 666–669.CrossRefGoogle Scholar
[18]Figà-Talamanca, A., and Gaudry, G. I., ‘Density and representation theorems for multipliers of type (p, q)’. To appear J. Austr. Math. Soc.Google Scholar
[19]Gaudry, G. I., ‘Quasimeasures and operators commuting with convolution’. To appear Pacific J. Math.Google Scholar
[20]Helgason, S., ‘Some problems in the theory of almost periodic functions’. Math. Scand. 3 (1955), 49–67.Google Scholar
[21]Helgason, S., ‘Multipliers of Banach algebras’. Ann. of Math. (2) 64 (1956), 240, 254.CrossRefGoogle Scholar
[22]Hörmander, L., Linear partial differential operators. Springer Verlag, Berlin (1963).Google Scholar
[23]Hörmander, L., ‘Estimates for translation-invariant operators in LD spaces’. Acta Math. 104 (1960), 93–140.Google Scholar
[24]Iwahori, N., ‘A proof of Tannaka duality theorem’. Sci. Papers Coll. Gen. Ed. Univ. Tokyo 8 (1958), 1–4.Google Scholar
[25]Johnson, B. E., ‘Continuity of transformations which leave invariant certain translation-invariant subspaces’. To appear Pacific J. Math.Google Scholar
[26]Kelley, J., ‘Averaging operators on C∞ (X)’. Illinois J. Math. 2 (1958), 214–223.CrossRefGoogle Scholar
[27]Larsen, R., Lin, T.-s and Wang, J.-k., ‘On functions with Fourier transfroms in Lp’. Michigan Math. J. 11 (1964), 367–378.Google Scholar
[28]Littman, W., ‘Multipliers in Lp and interpolation’, Bull. Amer. Math. Soc. 71 (1965), 764–766.Google Scholar
[29]Reiter, H., ‘Investigations in harmonic analysis’. Trans. Amer. Math. Soc. 73 (1952), 401–427.CrossRefGoogle Scholar
[30]Reiter, H., ‘Une propriété analytique d'une certaine classe de groupes localement compacts’. C.R. Acad. Sci. Paris 254 (1962), 3627–3629.Google Scholar
[31]Rudin, W., Fourier analysis on groups. Interscience Publishers, New York (1962).Google Scholar
[32]Schwartz, L., Théorie des distributions, I. Hermann, Paris (1950).Google Scholar
[33]Schwartz, L., Théorie des distributions, II. Hermann, Paris (1951).Google Scholar
[34]Stein, E. M., ‘On limits of sequences of operators’. Ann. of Math. 74 (1961), 140–170.Google Scholar
[35]Weil, A., L'Intégration dans les groupes topologiques et ses applications. Hermann, Paris (1953).Google Scholar
[36]Wendel, J. G., ‘On isometric isomorphisms of group algebras’. Pacific J. Math. 1 (1951), 305–311.CrossRefGoogle Scholar
[37]Wendel, J. G., ‘Left centralizers and isomorphisms of group algebras’. Pacific J. Math. 2 (1952), 251–261.Google Scholar
[38]Weston, J. D., ‘Positive perfect operators’. Proc. London Math. Soc. (3) 10 (1960), 545–565.Google Scholar
[39]Zygmund, A., Trigonometric series. Vols. I & II. Cambridge University Press (1959).Google Scholar