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A Note on Nilpotent Operators
Department of Mathematics, Duquesne University, Pittsburgh, USA
- 1 Department of Mathematics, Duquesne University, Pittsburgh, USA
Advances in Pure Mathematics·Volume 02 (2012)·Pages 367–370·Published 9 November 2012·DOI10.4236/apm.2012.26054
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Abstract
We find that a bounded linear operator T on a complex Hilbert space H satisfies the norm relation |||T| n a|| =2q, for any vector a in H such that q≤（||Ta||-4 -1 ||Ta|| 2 )≤1.A partial converse to Theorem 1 by Haagerup and Harpe in [1] is suggested. We establish an upper bound for the numerical radius of nilpotent operators.
KeywordsNumerical RangeNumerical RadiusNilpotent Operator Weighted ShiftEigenvalues
- U. Haagerup and P. de la Harpe, “The Numerical Radius of a Nilpotent Operator on a Hilbert Space,” Proceedings of the American Mathematical Society, Vol. 115, 1999, pp. 371-379.
- C. A. Berger and J. G. Stampi, “Mapping Theorems for the Numerical Range,” American Journal of Mathematics, Vol. 89, 1967, pp. 1047-1055.
- M. T. Karev, “The Numerical Range of a Nilpotent Operator on a Hilbert Space,” Proceedings of the American Mathematical Society, Vol. 132, 2004, pp. 2321-2326.
- J. P. Williams and T. Crimmins, “On the Numerical Radius of a Linear Operator,” American Mathematical Monthly, Vol. 74, No. 7, 1967, pp. 832-833. doi:10.2307/2315808
- J. T. Scheick, “Linear Algebra with Applications,” International Series in Pure and Applied Mathematics, Mc- Graw-Hill, New York, 1997.
- B. Sz.-Nagy and C. Folias, “Harmonic Analysis of Operators on a Hilbert Sapce,” North Holland, Amsterdam, 1970.