Numerical radius inequalities for operator matrices. (English) Zbl 06946450

Sahoo, Satyajit; Das, Namita; Mishra, Debasisha

Adv. Oper. Theory 4, No. 1, 197-214 (2019).

Summary: Several numerical radius inequalities for operator matrices are proved by generalizing earlier inequalities. In particular, the following inequalities are obtained: if \(n\) is even, \[ 2w(T) \leq \max\{\| A_1 \|, \| A_2 \|,\dots,\| A_n \|\}+\frac{1}{2}\sum_{k=0}^{n-1} \| |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|, \] and if \(n\) is odd, \[ 2w(T) \leq \max\{\| A_1 \|,\| A_2 \|,\dots,\| A_n \|\}+w\bigg(\widetilde{A}_{(\frac{n+1}{2})t}\bigg)+ \frac{1}{2}\sum_{k=0}^{n-1} \| |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|, \] for all \(t\in [0,1]\), \( A_i\)’s are bounded linear operators on the Hilbert space \(\mathcal{H}\), and \(T\) is off diagonal matrix with entries \(A_1, \cdots, A_n\).

MSC:

47A12	Numerical range, numerical radius
47A63	Linear operator inequalities
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

Aluthge transform; spectral radius; numerical radius; operator matrix; polar decomposition

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