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Search: MSC category 30J10 ( Blaschke products )

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1. CMB Online first

Reijonen, Atte
Remark on integral means of derivatives of Blaschke products
If $B$ is the Blachke product with zeros $\{z_n\}$, then $|B'(z)|\le \Psi_B(z)$, where $$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$ Moreover, it is a well-known fact that, for $0\lt p\lt \infty$, $$M_p(r,B')= \left(\frac{1}{2\pi}\int_{0}^{2\pi} |B'(re^{i\t})|^p\,d\t \right)^{1/p}, \quad 0\le r\lt 1,$$ is bounded if and only if $M_p(r,\Psi_B)$ is bounded. We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and $M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$. In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a Carleson-Newman Blaschke product and a weight $\omega$ satisfies a certain regularity condition, then $$ \int_\mathbb{D} |B'(z)|^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z), $$ where $dA(z)$ is the Lebesgue area measure on the unit disc.

Keywords:Bergman space, Blaschke product, Hardy space, integral mean
Categories:30J10, 30H10, 30H20

2. CMB 2012 (vol 57 pp. 80)

Khemphet, Anchalee; Peters, Justin R.
Semicrossed Products of the Disk Algebra and the Jacobson Radical
We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Keywords:semicrossed product, disk algebra, Jacobson radical
Categories:47L65, 47L20, 30J10, 30H50

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