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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.)
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