Spherically quasinormal tuples: $n$-th root problem and hereditary properties
Hranislav Stanković
TL;DR
This paper develops a multivariable framework for spherically quasinormal tuples by tying their characterization to the transform $${\Theta_{\mathbf{T}}}$$, their powers, and the spherical polar decomposition. It proves several equivalent conditions for spherical quasinormality, including $${\Theta_{\mathbf{T}}^{n}(I)=[{\Theta_{\mathbf{T}}(I)}]}^{n}$$$ for all $n$ and a spectral measure on $${\mathbb{R}}_{+}$$, and shows that powers preserve the property. A key result shows that a subnormal tuple with a spherical $n$-th root is spherically quasinormal, resolving a multivariable Curto et al. problem. The paper also links a pure spherically quasinormal tuple to its minimal normal extension and dual, establishing hereditary polar decomposition, purity of the dual, and that ${\mathbf{N}}$ is Taylor invertible iff ${\mathbf{T}}$ and ${\mathbf{S}}$ are left Taylor invertible.
Abstract
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 $Θ_{\mathbf{T}}(I)$. 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.