Non cyclic functions in the Hardy space of the bidisc with arbitrary decrease

TL;DR

This paper demonstrates that in the Hardy space of the bidisc, slow decrease of a function's modulus does not guarantee cyclicity, contrasting with the Bergman space case and extending known results from the disc.

Contribution

It provides a counterexample showing that slow decrease conditions are insufficient for cyclicity in the Hardy space of the bidisc, highlighting a key difference from the Bergman space.

Findings

Slow decrease of modulus does not imply cyclicity in the Hardy space of the bidisc

Counterexample illustrating non-cyclicity despite slow decrease

Contrast with known results in the Bergman space

Abstract

We construct an example to show that no condition of slow decrease of the modulus of a function is sufficient to make it cyclic in the Hardy space of the bidisc. This is similar to what is well known in the case of the Hardy space of the disc, but in contrast to the case of the Bergman space of the disc.