On some inequalities for the generalized joint numerical radius of semi-Hilbert space operators

CONDE, CRISTIAN; FEKI, KAIS

RICERCHE DI MATEMATICA

Springer

Año: 2024

0035-5038

Let $A$ be a positive (semidefinite) bounded linear operator on a complex Hilbert space $ig(mathcal{H}, langle cdot, cdotangle ig)$. The semi-inner product induced by $A$ is defined by ${langle x, yangle}_A := langle Ax, yangle$ for all $x, yinmathcal{H}$ and defines a seminorm ${|cdot|}_A$ on $mathcal{H}$. This makes $mathcal{H}$ into a semi-Hilbert space. For $pin [1,+infty )$, the generalized $A$-joint numerical radius of a $d$-tuple of operators $mathbf{T}=(T_1,ldots,T_d)$ is given byegin{align*}omega_{A,p}(mathbf{T})&=displaystylesup_{|x|_A=1}left(displaystylesum_{k=1}^d|iglangle T_kx, xigangle_A|^pight)^{rac{1}{p}}.end{align*}Our aim in this paper is to establish several bounds involving $omega_{A,p}(cdot)$. In particular, under suitable conditions on the operators tuple $mathbf{T}$ we generalize the well-known inequalities due to Kittaneh [Studia Math. 168 (2005), no. 1, 73--80].