Joint $\mathcal{A}$-operator seminorm bounds and refined joint $\mathcal{A}$-numerical radius inequalities

In this paper, we establish a new inequality relating the joint $\mathcal{A}$-operator seminorm of an $s$-tuple of operators $\mathbf{T}=({\mathcal{T}}_1,\dots ,{\mathcal{T}}_s)$, where each ${\mathcal{T}}_k$ admits an $\mathcal{A}$-adjoint, to its joint $\mathcal{A}$-Euclidean operator seminorm. The obtained bound is expressed in terms of the $\mathcal{A}$-numerical radius of suitable operator products involving the components of the tuple and their $\mathcal{A}$-adjoints. Here, $\mathcal{A}$ is a nonzero positive operator on a complex Hilbert space $\mathcal{H}$ inducing a semi-Hilbertian structure, and $s$ is a positive integer. We further derive refined upper and lower bounds for the joint $\mathcal{A}$-numerical radius of $\mathbf{T}$, denoted by ${\omega }_{\mathcal{A}}(\mathbf{T})$, thereby improving its classical equivalence with the joint $\mathcal{A}$-operator seminorm. In particular, the estimate $\frac{1}{2\sqrt{s}}\Vert \mathbf{T}{\Vert }_{\mathcal{A}}\le {\omega }_{\mathcal{A}}(\mathbf{T})\le \Vert \mathbf{T}{\Vert }_{\mathcal{A}}$ is sharpened by incorporating a term involving the $\mathcal{A}$-joint Crawford number of ${\mathbf{T}}^2$. Furthermore, we introduce a functional extension of the $\mathcal{A}$-joint numerical radius associated with a continuous nondecreasing function $\varphi$ satisfying $\varphi (0)=0$, and establish several inequalities connecting this extension with the classical $\mathcal{A}$-joint numerical radius and the $\mathcal{A}$-operator seminorm. The main contribution of this work is the use of the $\mathcal{A}$-adjoint framework in the study of multivariable operators within semi-Hilbertian spaces. Our approach combines algebraic operator methods with nonlinear functional techniques, leading to strict improvements of the classical equivalence bounds through the incorporation of the $\mathcal{A}$-joint Crawford number.