Improved inequalities for numerical radius via cartesian decomposition
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 1, pp. 24-37
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We derive various lower bounds for the numerical radius $w(A)$
of a bounded linear operator $A$ defined on a complex Hilbert space, which improve the existing
inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for $r\geq 1$, we show that
$$
\tfrac{1}{4}\|A^*A+AA^*\|\leq\tfrac{1}{2}(\tfrac{1}{2}\|\operatorname{Re}(A)+\operatorname{Im}(A)\|^{2r}+\tfrac{1}{2}\|\operatorname{Re}(A)-\operatorname{Im}(A)\|^{2r})^{1/r}
\leq w^{2}(A),
$$
where $\operatorname{Re}(A)$ and $\operatorname{Im}(A)$ are the real and imaginary parts of $A$, respectively.
Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper estimate
$w^2(A)\leq \frac{1}{2}(w(A^2)+\|A\|^2)$. Criteria for
$w(A)=\frac12\|A\|$ and for $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also given.
Keywords:
numerical radius, operator norm, Cartesian decomposition, bounded linear operator.
@article{FAA_2023_57_1_a1,
author = {P. Bhunia and S. Jana and M. S. Moslehian and K. Paul},
title = {Improved inequalities for numerical radius via cartesian decomposition},
journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
pages = {24--37},
publisher = {mathdoc},
volume = {57},
number = {1},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FAA_2023_57_1_a1/}
}
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P. Bhunia; S. Jana; M. S. Moslehian; K. Paul. Improved inequalities for numerical radius via cartesian decomposition. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 1, pp. 24-37. http://geodesic.mathdoc.fr/item/FAA_2023_57_1_a1/