Nayak's theorem for compact operators

Abstract

Let $A$ be an $m\times m$ complex matrix and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$ be the eigenvalues of $A$ arranged such that $|{\lambda }_1|\ge |{\lambda }_2|\ge \cdots \ge |{\lambda }_m|$ and for $n\ge 1,$ let $s_1^{(n)}\ge s_2^{(n)}\ge \cdots \ge s_m^{(n)}$ be the singular values of $A^n$. Then a famous theorem of Yamamoto (1967) states that $$\underset{n\to \infty }{\lim }(s_j^{(n)}{)}^{\frac{1}{n}}=|{\lambda }_j|,\forall 1\le j\le m.$$ 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 $|{\lambda }_1|,|{\lambda }_2|,$ $\dots ,|{\lambda }_m|.$ 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.

14 Pages