A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications

V. V. Peller

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 465-494

Résumé

The main result is the following description of Hankel operators in the Schatten-von Neumann class $\mathfrak{S}_p$ when $0$: $$ \Gamma_\varphi\in\mathfrak S_p\Leftrightarrow\varphi\in B_p^{1/p}, $$ where $\Gamma_\varphi$ is the Hankel operator with symbol $\varphi$, and $B_p^{1/p}$ is the Besov class. This result extends results obtained earlier for $1\leqslant p+\infty$ by the author to the case $ 0$. Also described are the Hankel operators in the Schatten–Lorentz classes $\mathfrak S_{pq}$, $0$, $ 0$. Precise descriptions of classes of functions defined in terms of rational approximation in the bounded mean oscillation norm are given as an application, along with a complete investigation of the case where the decrease is of power order, and some precise results on rational approximation in the $L^\infty$-norm. Certain other applications are also considered. Bibliography: 57 titles.