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

Authors : Kais Feki

Pages : 22-39

Doi:10.15672/hujms.1142554

View : 324 | Download : 667

Publication Date : 2024-02-29

Article Type : Research Paper

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.
Keywords : Semi inner product, positive operator, positive operator, A joint numerical radius, A Davis Wielandt radius, inequality