Compact operators with power-like asymptotics of singular numbers

M. Sh. Birman ; M. Z. Solomyak

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 21-30

Résumé

For a compact operator $A$ $(A\in\Upsilon_\infty)$ in a Hilbert space let $s_n(A)$, $n=1,2,\dots$, be the singular numbers of $A$ and $N(s; A)=\operatorname{card} \{n\in\mathbb N: s_n(A)>s\}$, $s>0$. Denote, for $0$ \begin{gather*} \Sigma_p=\{A\in\Upsilon_\infty: N(s, A)=O(s^{-p}), s\to0\},\\ \Sigma_p^0=\{A\in\Sigma_p: N(s, A)=o(s^{-p})\},\quad\sigma_p=\Sigma_p\setminus\Sigma_p^0. \end{gather*} The functionals $\Delta_p(A)=\limsup s^pN(s; A)$, $\delta_p(A)=\liminf s^pN(s; A)$, $s\to0$, finite for $A\in\Sigma_p$, depend on the class $a\in\sigma_p$ and not on an individual operator $A\in a$ (H. Weyl's lemma). So we may write $\Delta_p(a)$, $\delta_p(a)$, $a\in\sigma_p$. Some results for the functionals $\Delta_p$, $\delta_p$ (and similar functionals for positive and negative eigenvalues in the case $a=a^*=\{A^*:A\in a\}$) are obtained. In particular: I. For $a_1, a_2\in\sigma_p$ $[\Delta_p(a_1+a_2)]^{\frac1{p+1}}\leqslant[\Delta_p(a_1)]^{\frac1{p+1}}+[\Delta_p(a_2)]^{\frac1{p+1}}$. II. Let $a_1, a_2\in\sigma_p$, $a_1^*a_2=a_1a_2^*=0$, $\delta_p(a_i)=\Delta_p(a_i)$, $i=1, 2$. Then $\delta_p(a_1+a_2)=\Delta_p(a_1+a_2)=\Delta_p(a_1)+\Delta_p(a_2)$.