The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator A on $\mathcal{H}$. Here A is not assumed to be invertible. Mainly, if we denote by ${\omega }_A(\cdot )$ and $\omega (\cdot )$ the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ${\omega }_A(T)=\omega (A^{1/2}T(A^{1/2}{)}^{†}),$ where $(A^{1/2}{)}^{†}$ is the Moore-Penrose inverse of $A^{1/2}$. In addition, several new inequalities involving ${\omega }_A(\cdot )$ for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496].

本文涉及作用于复 Hilbert 空间的线性算子的广义数值半径$\mathcal{H}$，它们相对于由正算子A在$\mathcal{H}$. 这里不假定A是可逆的。主要是，如果我们用${\omega }_A(\cdot )$和$\omega (\cdot )$分别是广义和经典数值半径，我们证明对于每个A有界算子T我们有${\omega }_A(吨)=\omega (A^{\mathrm{1个}/\mathrm{2个}}吨(A^{\mathrm{1个}/\mathrm{2个}}{)}^{†}),$在哪里$(A^{\mathrm{1个}/\mathrm{2个}}{)}^{†}$是 Moore-Penrose 的逆函数$A^{\mathrm{1个}/\mathrm{2个}}$. 此外，一些新的不等式涉及${\omega }_A(\cdot )$为单个和多个操作员建立。特别是，通过使用新技术，我们涵盖并改进了 Najafi [线性代数应用。2020；588：489–496]。