# The Smirnov class for spaces with the complete Pick property

## TL;DR

This paper extends a theorem about functions in reproducing kernel Hilbert spaces with the complete Pick property, exploring implications for zero sets, space structures, and the corona problem.

## Contribution

It generalizes a known theorem to all such spaces, providing new insights into their structure and properties, including a counterexample to the corona theorem.

## Key findings

- Every function in the space is a quotient of multipliers.
- The paper discusses zero sets and Gleason parts in these spaces.
- Constructs a space where the corona theorem fails.

## Abstract

We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptano\u{g}lu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.