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

DOI10.1016/J.LAA.2021.08.014arXiv2105.09715MaRDI QIDQ6368107

Kallol Paul, Pintu Bhunia

Publication date: 20 May 2021

Abstract: Let $A$ be a bounded linear operator on a complex Hilbert space and $Re(A)$ ( $Im(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that �egin{eqnarray*} w(A)&geq &frac{1}{2} left |A ight| + frac{ 1}{2} mid |Re(A)|-|Im(A)|mid,,,mbox{and}\ w^2(A)&geq& frac{1}{4} left |A^*A+AA^* ight| + frac{1}{2}mid |Re(A)|^2-|Im(A)|^2 mid, end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=frac12sqrt|A^{*}A+A{A}^{*}|$ and prove that $w(A)=frac12sqrt|A^{*}A+A{A}^{*}|$ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $frac12sqrt|A^{*}A+A{A}^{*}|$. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.

Mathematics Subject Classification ID

Norms (inequalities, more than one norm, etc.) of linear operators (47A30) Norms of matrices, numerical range, applications of functional analysis to matrix theory (15A60) Numerical range, numerical radius (47A12)

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