Joint normality of operators in semi-Hilbertian spaces

In this paper, we introduce the concept of normality of a d-tuple of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$ when an additional semi-inner product induced by a positive operator A is considered. We investigate this new class of operators which is called A-normal d-tuple of operators. Mainly, we study the tensor product and the tensor sum of two A-normal d-tuples of operators. Also, we define the A-joint spectral radius associated with a d-tuple $\mathbf{T}=(T_1,\dots ,T_d)$ denoted $r_A\left(\mathbf{T}\right)$ and we show that under some conditions we have $r_A\left(\mathbf{T}\right)=|||\mathbf{T}||{|}_A$, where $|||\mathbf{T}||{|}_A$ denotes the A-joint operator seminorm of $\mathbf{T}$.