Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences