Nayak's theorem for compact operators

Abstract

Let $A$ be an $m\times m$ complex matrix and let $λ_1, λ_2, \ldots , λ_m$ be the eigenvalues of $A$ arranged such that $|λ_1|\geq |λ_2|\geq \cdots \geq |λ_m|$ and for $n\geq 1,$ let $s^{(n)}_1\geq s^{(n)}_2\geq \cdots \geq s^{(n)}_m$ be the singular values of $A^n$. Then a famous theorem of Yamamoto (1967) states that $$\lim _{n\to \infty}(s^{(n)}_j )^{\frac{1}{n}}= |λ_j|, ~~\forall \,1\leq j\leq m.$$ Recently S. Nayak strengthened this result very significantly by showing that the sequence of matrices $|A^n|^{\frac{1}{n}}$ itself converges to a positive matrix $B$ whose eigenvalues are $|λ_1|,|λ_2|,$ $\ldots , |λ_m|.$ Here this theorem has been extended to arbitrary compact operators on infinite dimensional complex separable Hilbert spaces. The proof makes use of Nayak's theorem, Stone-Weirstrass theorem, Borel-Caratheodory theorem and some technical results of Anselone and Palmer on collectively compact operators. Simple examples show that the result does not hold for general bounded operators.