Development of inequalities and characterization of equality conditions for the numerical radius

By Pintu Bhunia, Kallol Paul. 2021

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Abstract Let A be a bounded linear operator on a complex Hilbert space and ℜ ( A ) ( ℑ ( A ) ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we prove that w ( A ) ≥ 1 2 ‖ A ‖ + 1 2 | ‖ ℜ ( A ) ‖ − ‖ ℑ ( A ) ‖ | and w 2 ( A ) ≥ 1 4 ‖ A ⁎ A + A A ⁎ ‖ + 1 2 | ‖ ℜ ( A ) ‖ 2 − ‖ ℑ ( A ) ‖ 2 | , where w ( A ) and ‖ A ‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for w ( A ) = 1 2 ‖ A ⁎ A + A A ⁎ ‖ and prove that w ( A ) = 1 2 ‖ A ⁎ A + A A ⁎ ‖ if and only if the numerical range of A is a circular disk with center at the origin and radius 1 2 ‖ A ⁎ A + A A ⁎ ‖ . We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a4d2f8334fcade73bf5d972e7a72eb20976fec28