Author(s)

Abhay K. Gaur

Department of Mathematics, Duquesne University, Pittsburgh, USA.

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.

Numerical Range; Numerical Radius; Nilpotent Operator Weighted Shift; Eigenvalues

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.