A stronger form of Yamamoto's theorem II -- Spectral operators (2410.16318v2)

Abstract: For $A \in M_m(\mathbb{C})$, Yamamoto's generalization of the spectral radius formula (1967) asserts that $\lim_{n \to \infty} s_j(A^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$, where $s_j (A^n)$ denotes the $j^{\textrm{th}}$-largest singular value of $A^n$. Let $\mathscr{H}$ be a complex Hilbert space, and for an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, ${ |A^n|^{\frac{1}{n}} }_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove a spatial form of Yamamoto's theorem which asserts that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ 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 $II_1$ factor.