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IMPROVED INEQUALITIES FOR THE NUMERICAL RADIUS: WHEN INVERSE COMMUTES WITH THE NORM
Published online by Cambridge University Press: 02 February 2018
Abstract
New inequalities relating the norm $n(X)$ and the numerical radius
$w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that
$w(AB)\leq$
$2w(A)w(B)$ for invertible bounded linear Hilbert space operators
$A$ and
$B$. We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form
$n(X^{-1})=n(X)^{-1}$. We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example,
$w(AB)=w(A)w(B)$.
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- Research Article
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- © 2018 Australian Mathematical Publishing Association Inc.
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