Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T18:00:37.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Automatic discontinuity of intertwining operators

Published online by Cambridge University Press:  20 January 2009

Sandy Grabiner
Sandy Grabiner
Affiliation:
Department of MathematicsPomona CollegeClaremont, California 91711, U.S.A.
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.

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 49 - 54
Copyright
Copyright © Edinburgh Mathematical Society 1982

References

REFERENCES

1.Grabiner, S., Ranges of Products of Operators, Canadian J. Math. 26 (1974), 1430–1441.CrossRefGoogle Scholar
2.Grabiner, S., Operator Ranges and Invariant Subspaces, Indiana U. Math. J. 28 (1979), 845–857.CrossRefGoogle Scholar
3.Hilton, P., Lectures in Homological Algebra (C.B.M.S. Regional Conference Series in Mathematics, 8, American Math. Soc., Providence, R. I., 1971).CrossRefGoogle Scholar
4.Johnson, B. E., Continuity of Linear Operators Commuting with Continuous Linear Operators, Trans. Amer. Math. Soc. 128 (1967), 88–102.CrossRefGoogle Scholar
5.Johnson, B. E., Continuity of Operators Commuting with Quasi-nilpotent Operators, Indiana U. Math. J. 20 (1971), 913–915.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 (1968), 533–540.CrossRefGoogle Scholar
7.Kaplansky, I., Infinite Abelian Groups, Revised Edition (U. of Michigan Press, Ann Arbor, 1969).Google Scholar
8.Lay, D. C., Spectral Analysis Using Ascent, Descent, Nullity and Defect, Math. Ann. 184 (1970), 197–214.CrossRefGoogle Scholar
9.Newman, M. H. A., Topology of Plane Sets of Points, Second Edition (Cambridge U. Press, 1951).Google Scholar
10.Sinclair, A. M., A Discontinuous Intertwining Operator, Trans. Amer. Math. Soc. 188 (1974), 259–267.CrossRefGoogle Scholar
11.Sinclair, A. M., Automatic Continuity of Linear Operators (London Math. Soc. Lecture Notes, 21, Cambridge U. Press, 1976).CrossRefGoogle Scholar
12.Thomas, M. P., The Algebraic Structure of a Continuous Linear Operator on a Fréchet Space, preprint.Google Scholar