Abstract
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
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.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 24–37 https://doi.org/10.4213/faa3990.
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Bhunia, P., Jana, S., Moslehian, M.S. et al. Improved Inequalities for Numerical Radius via Cartesian Decomposition. Funct Anal Its Appl 57, 18–28 (2023). https://doi.org/10.1134/S0016266323010021
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DOI: https://doi.org/10.1134/S0016266323010021