Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T13:04:12.058Z Has data issue: false hasContentIssue false
  • English
  • Français

Translation-invariant linear operators

Published online by Cambridge University Press:  24 October 2008

H. G. Dales  and
A. Millinoton
H. G. Dales
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT
A. Millinoton
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT
Get access Rights & Permissions [Opens in a new window]

Extract

The theory of translation-invariant operators on various spaces of functions (or measures or distributions) is a well-trodden field. The problem is to decide, first, whether or not a linear operator between two function spaces on, say, ℝ or ℝ+ which commutes with one or many translations on the two spaces is necessarily continuous, and, second, to give a canonical form for all such continuous operators. In some cases each such operator is zero. The second problem is essentially the ‘multiplier problem’, and it has been extensively discussed; see [7], for example.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 161 - 172
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Albrecht, E. and Neumann, M. M.. Automatische Stetigkeitseigenschaften einiger Kiassen linear Operatoren. Math. Ann. 240 (1979), 251–280.CrossRefGoogle Scholar
[2]Albrecht, E. and Neumann, M. M.. Automatic continuity of local linear operators. Manuscripta Math. 32 (1980), 263–294.CrossRefGoogle Scholar
[3]Dunford, N. and Schwartz, J. T.. Linear Operators, part 1 (Interscience, 1967).Google Scholar
[4]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, vol. 2 (Springer-Verlag, 1970).Google Scholar
[5]Johnson, B. E.. Continuity of linear operators commuting with continuous linear operators. Trans. Amer. Math. Soc. 128 (1967), 88–102.CrossRefGoogle Scholar
[6]Johnson, B. E. and Sinclair, A. M.. Continuity of linear operators commuting with continuous linear operators, II. Trans. Amer. Math. Soc. 146 (1969), 533–540.CrossRefGoogle Scholar
[7]Larson, R.. An Introduction to the Theory of Multipliers (Springer-Verlag, 1971).CrossRefGoogle Scholar
[8]Laursen, K. B. and Neumann, M. M.. Decomposable operators and automatic continuity. J. Operator Theory 15, (1986), 33–51.Google Scholar
[9]Laursen, K. B. and Neumann, M. M.. Automatic continuity of intertwining linear operators on Banach spaces. Rend. Circ. Mat. Palermo 40 (1991), 325–341.CrossRefGoogle Scholar
[10]Loy, R. J.. Continuity of linear operators commuting with shifts. J. Funct. Anal. 16 (1974), 48–60.CrossRefGoogle Scholar
[11]Millington, A.. Linear operators which commute with translation. Thesis, University of Leeds (1991).Google Scholar
[12]Neumann, M. M. and Pták, V.. Automatic continuity, local type and causality. Studia Math. 82 (1985), 61–90.CrossRefGoogle Scholar
[13]Rudin, W.. Real and Complex Analysis, 3rd edition (McGraw-Hill, 1987).Google Scholar
[14]Sinclair, A. M.. A discontinuous intertwining operator. Trans. Amer. Math. Soc. 188 (1974), 259–267.CrossRefGoogle Scholar