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Some inequalities for normal operators in Hilbert spaces. (English) Zbl 1135.47004
Let $$T$$ be a normal operator acting on a Hilbert space $$H$$. By the Cauchy-Schwarz inequality, $$| \langle T^2x,x\rangle| = | \langle Tx,T^*x\rangle| \leq \| Tx\| \,\| T^*x\| =\| Tx\| ^2$$ ($$x\in H$$). Looking for upper bounds for the difference $$\| Tx\|^2-|\langle T^2x,x\rangle|$$, the author establishes some reverse inequalities for the inequality above by employing some inequalities for vectors in $$H$$ due to Buzano, Dunkl-Williams, Hile, Goldstein-Ryff-Clarke, Dragomir-Sándor and Dragomir. See also the author’s subsequently published paper [S. S. Dragomir, Banach J. Math. Anal. 1, No. 2, 154–175 (2007; Zbl 1136.47006)].

##### MSC:
 47A12 Numerical range, numerical radius 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)