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

The paper studies spaces of holomorphic functions on the unit ball in several complex variables. These radially weighted Besov spaces require that the radial derivative of the function is square-integrable against a given radial measure. The multiplier algebra consists of functions that keep the space closed under multiplication. A function is cyclic if multiplying it by multipliers produces a dense set in the space. Some of these spaces have the complete Pick property, which simplifies cyclicity questions, but more general ones do not. In the general case, there can be cyclic multipliers that fail to be weak-star sequentially cyclic. For bounded holomorphic functions without zeros, the authors give a condition on the logarithm of the function that guarantees cyclicity and also makes the reciprocal invertible in an associated Smirnov-type class. The condition is expressed using weak-star sequential cyclicity. A practical tool is the comparison principle: if one function has smaller absolute value than another and the smaller one is weak-star sequentially cyclic, then the larger one is too. This helps check the condition in concrete cases and gives new information about cyclicity outside the complete Pick setting.