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Zbl 1142.47007
Dragomir, S.S.
New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. (English)
[J] Linear Algebra Appl. 428, No. 11-12, 2750-2760 (2008). ISSN 0024-3795

If the selfadjoint operator $A$ on a Hilbert space $H$ is such that $mI\le A\le MI$, where $0< m< M$, then the Kantorovich inequality says that $1\le\langle Ax, x\rangle\langle A^{-1} x,x\rangle\le (m+ M)^2/(4mM)$ for any unit vector $x$ in $H$. In this paper, the author uses Grüss-type inequalities, which he obtained before, and their operator versions to establish inequalities of Kantorovich type for the more general class of operators $A$ satisfying $\text{Re}[(A^*- \overline\alpha I)(\beta I- A)]\ge 0$. The Grüss-type inequalities refer to the ones which give upper bounds for $|\langle u,v\rangle-\langle u,e\rangle\langle e,v\rangle|$ for vectors $u$, $v$ and $e$ in $H$ with $\Vert e\Vert= 1$ and scalars $\alpha$, $\beta$, $\gamma$ and $\delta$ satisfying $\Vert u- ((\alpha+ \beta)/2)e\Vert\le |\beta-\alpha|/2$ and $\Vert v- ((\gamma+\delta)/2) e\Vert\le |\delta- \gamma|/2$. There are also established estimates for $\Vert A\Vert^2- w(A)^2$ and $w(A)^2- w(A^2)$ for the above class of $A$. Here, $w(A)$ denotes the numerical radius $\sup\{|\langle Ax, x\rangle|: x\in H$, $\Vert x\Vert= 1\}$ of $A$.
[Pei Yuan Wu (Hsinchu)]

MSC 2000:: *47A12 Numerical range of linear operators
47A30 Operator norms and inequalities
47A63 Operator inequalities, etc.

Keywords: Kantorovich inequality; Grüss inequality; numerical radius; bounded linear operators

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Zentralblatt MATH Berlin [Germany]