A Note on Nilpotent Operators

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.

We find that a bounded linear operator T on a complex Hilbert space

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.

A. Gaur, "A Note on Nilpotent Operators,"

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.

[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.