Numerical radius inequalities of $2 \times 2$ operator matrices

DOI10.1007/S43036-022-00237-7arXiv2105.09718MaRDI QIDQ6368108

Kallol Paul, Pintu Bhunia

Publication date: 20 May 2021

Abstract: Several upper and lower bounds for the numerical radius of $2imes2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then �egin{eqnarray*} && frac{1}{2}max left { |B|, |C| ight }+frac{1}{4} left | |B+C^*|-|B-C^*| ight | &&leq w left(left[�egin{array}{cc} 0 & B C& 0 end{array} ight] ight)\ &&leq frac{1}{2} max left{|B|,|C| ight }+frac{1}{2}max left {r^{frac{1}{2}}(|B||C^*|),r^{frac{1}{2}}(|B^*||C|) ight}, end{eqnarray*} where $w(.)$, $r(.)$ and $|.|$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $maxBig|B+C{|}^2,|B-C{|}^{2}Bigleqleft|B^2+C^{2}ight|+2w(|B||C|).$

Mathematics Subject Classification ID

Norms (inequalities, more than one norm, etc.) of linear operators (47A30) Numerical range, numerical radius (47A12) Operator matrices (47A08)

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