Some numerical radius inequalities for Hilbert space operators

Omidvar, Mohsen Erfanian; Sal Moslehian, Mohammad; Niknam, Asdolah

doi:10.2140/involve.2009.2.471

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Some numerical radius inequalities for Hilbert space operators

Mohsen Erfanian Omidvar, Mohammad Sal Moslehian and Assadollah Niknam

Vol. 2 (2009), No. 4, 471–478

DOI: 10.2140/involve.2009.2.471

Abstract

We present several numerical radius inequalities for Hilbert space operators. More precisely, we prove that if $A, B, C, D \in B (H)$ and $T = [\begin{matrix} A & B \\ C & D \end{matrix}]$ then $max (w (A), w (D)) \leq (1 ∕ 2) (∥ T ∥ + ∥ T^{2} ∥^{1 ∕ 2})$ and $max ({(w (B C))}^{1 ∕ 2}, {(w (C B))}^{1 ∕ 2}) \leq (1 ∕ 2) (∥ T ∥ + ∥ T^{2} ∥^{1 ∕ 2})$ . We also show that if $A \in B (H)$ is positive, then

w (A X - X A) \leq \frac{1}{2} ∥ A ∥ (∥ X ∥ + ∥ X^{2} ∥^{1 ∕ 2}) .

Keywords

bounded linear operator, Hilbert space, norm inequality, numerical radius, positive operator

Mathematical Subject Classification 2000

Primary: 47A62

Secondary: 46C15, 47A30, 15A24

Milestones

Received: 5 May 2009

Accepted: 1 July 2009

Published: 28 October 2009

Communicated by Kenneth S. Berenhaut

Authors

Mohsen Erfanian Omidvar
Department of Mathematics Faculty of Science Islamic Azad University-Mashhad Branch Mashhad 91722 Iran

Mohammad Sal Moslehian
Department of Pure Mathematics Center of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University of Mashhad P.O. Box 1159 Mashhad 91775 Iran
http://www.um.ac.ir/~moslehian
Assadollah Niknam
Department of Pure Mathematics Center of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University of Mashhad P.O. Box 1159 Mashhad 91775 Iran

Vol. 2, No. 4, 2009