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A Note on Nilpotent Operators

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DOI: 10.4236/apm.2012.26054    3,176 Downloads   5,872 Views  
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ABSTRACT

We find that a bounded linear operator T on a complex Hilbert space H satisfies the norm relation |||T|na|| =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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

A. Gaur, "A Note on Nilpotent Operators," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 367-370. doi: 10.4236/apm.2012.26054.

References

[1] 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.
[2] C. A. Berger and J. G. Stampi, “Mapping Theorems for the Numerical Range,” American Journal of Mathematics, Vol. 89, 1967, pp. 1047-1055.
[3] 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.
[4] 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
[5] J. T. Scheick, “Linear Algebra with Applications,” International Series in Pure and Applied Mathematics, Mc- Graw-Hill, New York, 1997.
[6] B. Sz.-Nagy and C. Folias, “Harmonic Analysis of Operators on a Hilbert Sapce,” North Holland, Amsterdam, 1970.

  
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