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On k-quasihyponormal operators II

Published online by Cambridge University Press:  17 April 2009

B.C. Gupta  and
P.B. Ramanujan
B.C. Gupta
Affiliation:
Department of Mathematics, Sadar Patel University, Vallabh Vidyanagar 388 120, Gujarat, India.
P.B. Ramanujan
Affiliation:
Department of Mathematics, Saurashtra University, Rajkot 360 005, Gujarat, India.
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Abstract

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An operator T on a Hilbert space is in the class of k-quasihyponormal operators Q(k), if T*k(T*T−TT*)Tk ≥ 0. It is shown that if T is in Q(k) and S is normal such that TX = XS, where X is one to one with dense range, then T is normal; and is unitarily equivalent to S. It is proved that S can be replaced by a cohyponormal operator, if T in Q(1) is one to one. It is also shown that two quasisimilar operators in Q(k) have equal spectra, and every reductive operator quasisimilar to a normal operator is normal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 1 , August 1981 , pp. 61 - 67
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Apostol, Constantin, “Operators quasisimilar to a normal operator”, Proc. Amer. Math. Soc. 53 (1975), 104–106.Google Scholar
[2]Campbell, Stephen L., “Linear operators for which T*T and T + T* commute III”, Pacific J. Math. 76 (1978), 17–19.Google Scholar
[3]Campbell, Stephen L., “Operator-valued inner functions analytic on the closed disc II”, Pacific J. Math. 60 (1975), no. 2, 37–49.Google Scholar
[4]Campbell, S.L. and Gupta, B.C., “On k-quasihyponormal operators”, Math. Japon. 23 (1978), 185–189.Google Scholar
[5]Clary, Stuart, “Equality of spectra of quasisimilar hyponormal operators”, Proc. Amer. Math. Soc. 53 (1975), 88–90.Google Scholar
[6]Nordgren, Eric, Radjavi, Heydar, and Rosenthal, Peter, “On operators with reducing invariant subspaces”, Amer. J. Math. 97 (1975), 559–570.Google Scholar
[7]Putnam, C.R., “Hyponormal contractions and strong power convergence”, Pacific J. Math. 57 (1975), 531–538.Google Scholar
[8]Radjabalipour, M., “Ranges of hyponormal operators”, Illinois J. Math. 21 (1977), 70–75.CrossRefGoogle Scholar
[9]Sheth, I.H., “On hyponormal operators”, Proc. Amer. Math. Soc. 17 (1966), 998–1000.Google Scholar
[10]Sheth, I.H., “Quasi-hyponormal operators”, Rev. Roumaine Math. Pures Appl. 19 (1976). 1049–1053.Google Scholar
[11]Stampfli, Joseph G. & Wadhwa, Bhushan L., “An asymmetric Putnam-Fugledge theorem for dominant operators”, Indiana Univ. Math. J. 25 (1976), 359–365.Google Scholar
[12]Williams, James P., “Operators similar to their adjoints”, Proc. Amer. Math. Soc. 20 (1969), 121–123.Google Scholar