I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play between functional analysis, operator theory and complex variables.
But what are the applications of this subject elsewhere?
Would the average professional operator theory person know much about this?
Does it come up in theoretical physics?
Bergman spaces have applications in problems of quantization. For details see, for example this paper Berezin and Berezin-Toeplitz Quantizations for General Function Spaces
Some general description of quantization from mathematical point of view is here Quantization Methods: A Guide for Physicists and Analysts