For a matrix $A \in M_m(\mathbb{C})$ and $1 \le j \le m$, let $ s_j(A^n) $ denote the $j^\mathrm{th}$ largest singular value of $A^n$. A classical result due to Yamamoto (1967) asserts that $\lim_{n \to \infty} s_j(A^n)^{\frac{1}{n}}$ is equal to the $j^\mathrm{th}$-largest eigenvalue-modulus of $A$, counted with multiplicity. Nayak (2023) generalized this result significantly by showing that for $A \in M_m(\mathbb C)$, the sequence ${|A^n|^\frac{1}{n}}$, where $|A|:=(A^*A)^\frac{1}{2}$, converges to a positive-semidefinite matrix, whose $j^\mathrm{th}$-largest eigenvalue is equal to the $j^\mathrm{th}$-largest eigenvalue-modulus of $A$, and in fact provided an explicit description of the limit.

In this talk, we will extend the Yamamoto–Nayak result to the context of spectral operators on a (infinite-dimensional) complex Hilbert space $\mathscr H$. Specifically, we will show that the normalized power sequence, ${ |A^n|^{\frac{1}{n}} }_{n \in \mathbb N}$, of a spectral operator $A \in \mathscr B(\mathscr H)$ converges in norm, and provide an explicit description of the limiting positive operator. This framework not only generalizes Nayak’s earlier result to infinite-dimensional settings, but also offers an alternative and streamlined proof of the finite-dimensional case. (Joint with S. Nayak.)