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  is suggested. We establish an upper bound for the numerical radius of nilpotent operators.
Cite this paper
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