Linear operators commuting with translations on $\mathcal {D}(\mathbf {R})$ are continuous
HTML articles powered by AMS MathViewer
- by Gary Hosler Meisters PDF
- Proc. Amer. Math. Soc. 106 (1989), 1079-1083 Request permission
Abstract:
Let $\mathcal {D}\left ( {\mathbf {R}} \right )$ denote the Schwartz space of all ${C^\infty }$-functions $f:{\mathbf {R}} \to {\mathbf {C}}$ with compact supports in the real line ${\mathbf {R}}$. An earlier result of the author on the automatic continuity of translation-invariant linear functionals on $\mathcal {D}\left ( {\mathbf {R}} \right )$ is combined with a general version of the Closed-Graph Theorem due to A. P. Robertson and W. J. Robertson in order to prove that every linear mapping $S$ of $\mathcal {D}\left ( {\mathbf {R}} \right )$ into itself, which commutes with translations, is automatically continuous.References
- Lilian Asam, Translation-invariant linear forms on ${\scr E}β(G)$ and non-Liouville tuples on $G$, J. Funct. Anal. 78 (1988), no.Β 1, 1β12. MR 937628, DOI 10.1016/0022-1236(88)90128-0
- John M. Bachar, Philip C. Curtis Jr., H. Garth Dales, and Marc P. Thomas (eds.), Radical Banach algebras and automatic continuity, Lecture Notes in Mathematics, vol. 975, Springer-Verlag, Berlin-New York, 1983. MR 697577
- Jean Bourgain, Translation invariant forms on $L^p(G)\ (1<p<\infty )$, Ann. Inst. Fourier (Grenoble) 36 (1986), no.Β 1, 97β104. MR 840715, DOI 10.5802/aif.1039
- H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), no.Β 2, 129β183. MR 500923, DOI 10.1112/blms/10.2.129 W. F. Donoghue, Jr., Distributions and Fourier transforms, Academic Press, New York, 1969.
- B. E. Johnson, Continuity of linear operators commuting with continuous linear operators, Trans. Amer. Math. Soc. 128 (1967), 88β102. MR 213894, DOI 10.1090/S0002-9947-1967-0213894-5
- Barry Edward Johnson, A proof of the translation invariant form conjecture for $L^{2}(G)$, Bull. Sci. Math. (2) 107 (1983), no.Β 3, 301β310. MR 719270
- B. E. Johnson and A. M. Sinclair, Continuity of linear operators commuting with continuous linear operators. II, Trans. Amer. Math. Soc. 146 (1969), 533β540. MR 251564, DOI 10.1090/S0002-9947-1969-0251564-X
- JarosΕaw Krawczyk, Continuity of operators commuting with translations for compact groups, Monatsh. Math. 107 (1989), no.Β 2, 125β130. MR 994978, DOI 10.1007/BF01300918
- Richard J. Loy, Continuity of linear operators commuting with shifts, J. Functional Analysis 17 (1974), 48β60. MR 0353046, DOI 10.1016/0022-1236(74)90003-2
- Gary H. Meisters, Translation-invariant linear forms and a formula for the Dirac measure, Bull. Amer. Math. Soc. 77 (1971), 120β122. MR 267397, DOI 10.1090/S0002-9904-1971-12627-7
- Gary H. Meisters, Translation-invariant linear forms and a formula for the Dirac measure. , J. Functional Analysis 8 (1971), 173β188. MR 0288575, DOI 10.1016/0022-1236(71)90025-5
- Gary Hosler Meisters, Some problems and results on translation-invariant linear forms, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981) Lecture Notes in Math., vol. 975, Springer, Berlin, 1983, pp.Β 423β444. MR 697605, DOI 10.1007/BFb0064574
- Alex P. Robertson and Wendy Robertson, On the closed graph theorem, Proc. Glasgow Math. Assoc. 3 (1956), 9β12. MR 84108, DOI 10.1017/S2040618500033372
- A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
- Joseph Rosenblatt, Translation-invariant linear forms on $L_p(G)$, Proc. Amer. Math. Soc. 94 (1985), no.Β 2, 226β228. MR 784168, DOI 10.1090/S0002-9939-1985-0784168-5
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-DΓΌsseldorf-Johannesburg, 1973. MR 0365062
- Sadahiro Saeki, Discontinuous translation invariant functionals, Trans. Amer. Math. Soc. 282 (1984), no.Β 1, 403β414. MR 728720, DOI 10.1090/S0002-9947-1984-0728720-5
- Allan M. Sinclair, Automatic continuity of linear operators, London Mathematical Society Lecture Note Series, No. 21, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0487371, DOI 10.1017/CBO9780511662355
- G. A. Willis, Translation invariant functionals on $L^p(G)$ when $G$ is not amenable, J. Austral. Math. Soc. Ser. A 41 (1986), no.Β 2, 237β250. MR 848037, DOI 10.1017/S1446788700033656
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 1079-1083
- MSC: Primary 47B38; Secondary 46F10
- DOI: https://doi.org/10.1090/S0002-9939-1989-0979227-1
- MathSciNet review: 979227