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, \ldots, \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 $ \mathscr{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}}\|\mathbf{T}\|_{\mathcal{A}} \leq \omega_{\mathcal{A}}(\mathbf{T}) \leq \|\mathbf{T}\|_{\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 $ \phi $ satisfying $ \phi(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.