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Zbl 1135.47004
Dragomir, S.S.
Some inequalities for normal operators in Hilbert spaces.
(English)
[J] Acta Math. Vietnam. 31, No. 3, 291-300 (2006). ISSN 0251-4184

Let $T$ be a normal operator acting on a Hilbert space $H$. By the Cauchy-Schwarz inequality, $\vert \langle T^2x,x\rangle\vert = \vert \langle Tx,T^*x\rangle\vert \leq \Vert Tx\Vert \,\Vert T^*x\Vert =\Vert Tx\Vert ^2$ ($x\in H$). Looking for upper bounds for the difference $\Vert Tx\Vert^2-\vert\langle T^2x,x\rangle\vert$, 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 [{\it S. S. Dragomir}, Banach J.~Math. Anal. 1, No. 2, 154--175 (2007; Zbl 1136.47006)].
MSC 2000:
*47A12 Numerical range of linear operators
47A05 General theory of linear operators

Keywords: numerical radius; bounded linear operator; normal operator; Hilbert space; norm inequality

Citations: Zbl 1136.47006

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