Originally Published In
Aleman, A., Hartz, M., McCarthy, J. E. and Richter, S. (2017), The Smirnov class for spaces with the complete Pick property. J. London Math. Soc., 96: 228–242. doi:10.1112/jlms.12060
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 Kaptanoğlu. 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.
Aleman, Alexandru; Hartz, Michael; McCarthy, John E.; and Richter, Stefan, "Smirnov class for spaces with the complete Pick property" (2017). Mathematics Faculty Publications. 44.