On the convergence of the normalized power sequence of spectral operators on Hilbert space

Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B}(\mathcal{H})$ denote the set of all bounded operators on $\mathcal{H}$ . For an operator $T\in \mathcal{B}(\mathcal{H})$, let $\mathrm{\mid }T\mathrm{\mid }:=(T^{\ast }T{)}^{\frac{1}{2}}$. For $A$ in $\mathcal{B}(\mathcal{H})$, we refer to the sequence, $\{\mathrm{\mid }{A}^n{\mathrm{\mid }}^{\frac{1}{n}}{\}}_{n\in \mathbb{N}}$, as the $\mathit{\text{normalized}}\mathit{\text{power}}\mathit{\text{sequence}}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathcal{B}(\mathcal{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $I{I}_1$ factor.