Spherically quasinormal tuples: $n$-th root problem and hereditary properties

In this paper, we provide several characterizations of a spherically quasinormal tuple $\mathbf{T}$ in terms of its normal extension, as well as in terms of powers of the associated elementary operator $\Theta_{\mathbf{T}}(I)$. Utilizing these results, we establish that the powers of spherically quasinormal tuples remain spherically quasinormal. Additionally, we prove that the subnormal $n$-roots of spherically quasinormal tuples must also be spherically quasinormal, thereby resolving a multivariable version of a previously posed problem by Curto et al. in [17]. Furthermore, we investigate the connection between a (pure) spherically quasinormal tuple $\mathbf{T}$, its minimal normal extension $\mathbf{N}$, and its dual $\mathbf{S}$. Among other things, we show that $\mathbf{T}$ inherits the spherical polar decomposition from $\mathbf{N}$. Finally, we also demonstrate that $\mathbf{N}$ is Taylor invertible if and only if $\mathbf{T}$ and $\mathbf{S}$ have closed ranges.