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Hyponormal and essentially normal operators†

Published online by Cambridge University Press:  14 November 2011

C. R. Putnam
C. R. Putnam
Affiliation:
Purdue University, West Lafayette, Indiana, U.S.A.
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Synopsis

Let T be a hyponormal operator on a Hilbert space, so that T*T – TT*≧ 0. Let T have the Cartesian representation T = H + iJ where H has the spectral family {Et} and suppose that EtJ − JEt is compact for almost all t on a Borei set α satisfying E(α) = I. The principal result (Theorem 3) is that under these hypotheses T must be normal. In case T is hyponormal and essentially normal some sufficient conditions are given assuring that, for a fixed t, EtJ − JEt is compact.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 123 - 132
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Aiken, J. G.. An application of direct integral theory to a question of Calkin. Notices Amer. Math. Soc. 21 (1974), A493.Google Scholar
2Berberian, S. K.. Lectures in functional analysis and operator theory (Heidelberg: Springer, 1974).CrossRefGoogle Scholar
3Brown, L. G., Douglas, R. G. and Fillmore, P. A.. Unitary equivalence modulo the compact operators and extensions of C*-algebras. Lecture Notes in Mathematics 145, 58–128 (Berlin: Springer, 1973).Google Scholar
4Calkin, J. W.. Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. of Math. 42 (1941), 839–873.CrossRefGoogle Scholar
5Clancey, K. F. and Putnam, C. R.. The spectra of hyponormal operators. Comment. Math. Helv. 46 (1971), 541–456.CrossRefGoogle Scholar
6Douglas, R. G.. Banach algebra techniques in operator theory (New York: Academic Press, 1972).Google Scholar
7Fillmore, P. A., Stampili, J. G. and Williams, J. P.. On the essential minimal range, the essential spectrum and a problem of Halmos. Acta Sci. Math. 33 (1972), 179–192.Google Scholar
8Kato, T.. Smooth operators and commutators. Studia Math. 31 (1968), 535–546.CrossRefGoogle Scholar
9Pincus, J. D.. Commutators and systems of singular integral equations, I. Acta Math. 121 (1968), 219–249.CrossRefGoogle Scholar
10Pincus, J. D.. The spectrum of seminormal operators. Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1684–1685.CrossRefGoogle ScholarPubMed
11Putnam, C. R.. Commutation properties of Hilbert space operators and related topics. Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 36 (New York: Springer, 1967).CrossRefGoogle Scholar
12Putnam, C. R.. An inequality for the area of hyponormal spectra. Math. Z. 116 (1970), 323–330.CrossRefGoogle Scholar
13Putnam, C. R.. A similarity between hyponormal and normal spectra. Illinois J. Math. 16 (1972), 695–702.CrossRefGoogle Scholar
14Putnam, C. R.. Hyponormal operators having real parts with simple spectra. Trans. Amer. Math. Soc. 172 (1972), 447–164.CrossRefGoogle Scholar
15Putnam, C. R.. A norm inequality in hyponormal operator theory. Michigan Math. J. 22 (1975), 195–200.Google Scholar
16Williams, J. P.. Diagonalizable normal operators. Proc. Amer. Math. Soc. 54 (1976), 106–108.CrossRefGoogle Scholar