Cyclicity via weak$^\ast$ sequentially cyclicity in Radially weighted Besov spaces

TL;DR

This paper investigates cyclicity properties of functions in radially weighted Besov spaces, establishing conditions for cyclicity and invertibility, especially in spaces lacking the complete Pick property.

Contribution

It introduces a new condition on $ ext{log} f$ for cyclicity in these spaces and explores the relationship between weak$^ ext{*}$ sequential cyclicity and multiplier comparison principles.

Findings

A condition on $ ext{log} f$ implies cyclicity and invertibility in Besov spaces.

Weak$^ ext{*}$ sequential cyclicity can be verified via a comparison principle.

Results extend understanding of cyclicity in spaces without the complete Pick property.

Abstract

A radially weighted Besov space $H$ is a space of holomorphic functions on the unit ball $\mathbb{B}_d \subseteq \mathbb{C}^d$ whose $N$-th radial derivative is square integrable with respect to a given admissible radial measure. We write $Mult(H)$ for its multiplier algebra. The cyclic vectors in $H$ are those functions $f$ whose multiplier multiples are dense in $H$. We call a multiplier has the complete Pick property. However, in more general radially weighted Besov spaces there may be multipliers that are cyclic, but not weak$^\ast$ sequentially cyclic. For bounded holomorphic functions $f$ with no zeros in $\mathbb{B}_d$, we obtain a condition on $\log f$ that implies the cyclicity of $f$ in $H$ and yields invertibility properties for $1/f$ within an associated Smirnov-type class. This condition is formulated in terms of weak$^\ast$ sequentially cyclic multipliers and can often be verified using a comparison principle: if $f, g \in Mult(H)$ satisfy $|f| \leq |g|$ and if $f$ is weak$^\ast$ sequentially cyclic, then $g$ is also weak$^\ast$ sequentially cyclic. These results provide new insights into cyclicity phenomena in radially weighted Besov spaces in settings, where $H$ fails to be a complete Pick space.