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

Let A be a positive (semidefinite) bounded linear operator on complex Hilbert space $$\big ({\mathcal {H}}, \langle \cdot , \rangle \big )$$ ( H ? · ? ) . The semi-inner product induced by is defined $${\langle x, y\rangle }_A := Ax, $$ x y : = for all $$x, y\in {\mathcal {H}}$$ ? and defines seminorm $${\Vert \Vert }_A$$ ? $${\mathcal This makes into semi-Hilbert space. For $$p\in [1,+\infty p [ 1 + ? the generalized A-joint numerical radius of d-tuple operators $${\mathbf {T}}=(T_1,\ldots ,T_d)$$ T … d given $$\begin{aligned} \omega _{A,p}({\mathbf {T}}) =\sup _{\Vert x\Vert _A=1}\left( \sum _{k=1}^d|\big T_kx, x\big _A|^p\right) ^{\frac{1}{p}}. \end{aligned}$$ ? sup ? k | ? ? Our aim in this paper to establish several bounds involving $$\omega _{A,p}(\cdot In particular, under suitable conditions tuple {T}}$$ we generalize well-known inequalities due Kittaneh (Studia Math 168(1):73–80, 2005).