Jointly $A$-hyponormal $m$-tuple of commuting operators and related results

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $A$ on a complex Hilbert space $\mathcal{X}$, which is called jointly $A$-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $m$-tuples of operators that admit adjoint operators with respect to $A$. Mainly, we prove that if $\mathbf{B}=(B_1,\cdots ,B_m)$ is a jointly $A$-hyponormal $m$-tuple of commuting operators, then $\mathbf{B}$ is jointly $A$-normaloid. This result allows us to establish, for a particular case when $A$ is the identity operator, a sharp bound for the distance between two jointly hyponormal $m$-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $A$- $p$-hyponormal operators with $0<p<1$. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.