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On the commutants modulo Cp of A2 and A3

Published online by Cambridge University Press:  09 April 2009

Fuad Kittaneh
Fuad Kittaneh
Affiliation:
Department of Mathematics, United Arab Emirates University, P. O. Box 15551 A1-Ain, U.A.E.
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Abstract

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We prove the following statements about bounded linear operators on a complex separable infinite dimensional Hilbert space. (1) Let A and B* be subnormal operators. If A2X = XB2 and A3X = XB3 for some operator X, then AX = XB. (2) Let A and B* be subnormal operators. If A2X – XB2 ∈ Cp and A3X – XB3 ∈ Cp for some operator X, then AX − XB ∈ C8p. (3) Let T be an operator such that 1 − T*T ∈ Cp for some p ≥1. If T2X − XT2 ∈ Cp and T3X – XT3 ∈ Cp for some operator X, then TX − XT ∈ Cp. (4) Let T be a semi-Fredholm operator with ind T < 0. If T2X − XT2 ∈ C2 and T3X − XT3 ∈ C2 for some operator X, then TX − XT ∈ C2.

Keywords

commutantnormal operatorssubnormal operatorsisometrySchatten p-class
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 47 - 50
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Al-Moajil, A. H., ‘The commutants of relatively prime powers in Banach algebras’, Proc. Amer. Math. Soc. 57 (1976), 243–249.CrossRefGoogle Scholar
[2]Embry, M. R., ‘nth roots of operators’, Proc. Amer. Math. Soc. 19 (1968), 63–68.Google Scholar
[3]Fillmore, P., Stampfli, J. and Williams, J., ‘On the essential numerical range, the essential spectrum, and a problem of Halmos’, Acta Sci. Math. (Szeged) 33 (1972), 179–192.Google Scholar
[4]Pearcy, C., Some recent developments in operator theory (Lecture Notes No. 36, Amer. Math. Soc., Providence, R. I., 1978).CrossRefGoogle Scholar
[5]Weiss, G., ‘The Fuglede commutativity theorem modulo operator ideals’, Proc. Amer. Math. Soc. 83 (1981), 113–118.Google Scholar