Closed ideals of algebras of type $B_{p,q}^\alpha$

N. A. Shirokov

Izvestiya. Mathematics , Tome 21 (1983) no. 3, pp. 585-600

Résumé

Let $B_{p,q}^\alpha$ be the space of functions analytic in the unit disk, with the norm $$ |f(0)|+\sup_{01}\Biggl[\int_0^\pi\frac{dh}{h^{1+\alpha q}} \biggl(\int_0^{2\pi}|f(re^{i(\theta+h)})-f(re^{i\theta})|^p\,d\theta\biggr)^{q/p}\Biggr]^{1/q}, $$ where $0\alpha1$, $p>1/\alpha$ and $1\leqslant q\leqslant\infty$, and let $C_A$ be the space of functions analytic in the unit disk and continuous in its closure. All closed ideal are described for spaces more general than $B_{p,q}^\alpha$; it is shown that for every closed ideal $I\subset B_{p,q}^\alpha$ there is a closed ideal $I_0\subset C_A$ such that $I=I_0\cap B_{p,q}^\alpha$, and conversely. Bibliography: 13 titles.