Hankel operators of class $\mathfrak S_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators)

V. V. Peller

Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 443-479

Résumé

A criterion is given for a Hankel operator $H_\varphi\colon H^2\to H^2_-$ ($H_\varphi f=(I-\mathbf P)\varphi f$, where $\mathbf P$ is the orthogonal projection of $L^2$ onto $H^2$) to belong to the Schatten–von Neumann class $\mathfrak S_p$ in terms of its symbol $\varphi$. Various applications are considered: a precise description is obtained for classes of functions definable in terms of rational approximation in the $BMO$ (bounded mean oscillation) norm; it is proved that the averaging projection onto the set of Hankel operators is bounded in the norm of $\mathfrak S_p$, $1$; a counterexample is given to a conjecture of Simon on the majorization property in $\mathfrak S_p$; a problem of Ibragimov and Solev on stationary Gaussian processes is solved; and a criterion is obtained for functions of an operator in the Sz.-Nagy–Foias model to belong to the class $\mathfrak S_p$. Bibliography: 47 titles.