Kleinecke–Shirokov theorem: a version for isometric transformations

In this paper, we present a version of the Kleinecke–Shirokov Theorem applicable to isometries on a Hilbert space \({\mathcal {H}}\). Specifically, we demonstrate that if \( V \in {\mathfrak {B}}({\mathcal {H}})\) is a quasinormal partial isometry and \(T \in {\mathfrak {B}}({\mathcal {H}})\) satisfies \({\mathcal {R}}(T) \subseteq {\mathcal {R}}(V)\), then

$$\begin{aligned} [V,[V,T]]=0\quad \implies \quad [V,T]=0. \end{aligned}$$

We also consider the mixed commutators of two isometries, and their belonging to the Schatten-von Neumann classes. Finally, we show that the corresponding classical statement regarding normal operators can be derived from our results.