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

Research paper by **Kais Feki**

Indexed on: **12 May '20**Published on: **10 May '20**Published in: **arXiv - Mathematics - Functional Analysis**

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#### Abstract

Let $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$ 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 ${\langle
x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ 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\mid x\rangle}_A\big|^2+\big|{\langle
Sx\mid x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several
bounds involving $\omega_{A,\text{e}}(T,S)$. Moreover, several inequalities
related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators is
established. Some of the obtained bounds generalize and refine some earlier
results of Zamani and Shebrawi [Mediterr. J. Math. 17, 25 (2020)].