Inequalities for the $A$-joint numerical radius of two operators and their applications

Abstract

Let $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ and defines a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} \omega_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx, x\rangle}_A\big|^2+\big|{\langle Sx, x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving $\omega_{A,\text{e}}(T,S)$. This allows us to establish some inequalities for the $A$-numerical radius of $A$-bounded operators. In particular, we extend the well-known inequalities due to Kittaneh [Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (1), 73-80, 2005]. Moreover, several bounds related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators are also provided.