This presentation, covering holomorphic function theory and multivariable operator theory, is divided into two parts.

In the first part, we discuss the interaction between two classical theorems: the Carathéodory approximation theorem and the Herglotz representation theorem, showing their equivalence in classical settings. We then extend the Carathéodory approximation to finitely connected domains for operator-valued functions. Then, an integral representation for Herglotz functions in finitely connected planar domains is derived in the style of Korányi and Pukánszky using the Carathéodory approximation as a tool.

Furthermore, we analyze operator-valued Herglotz functions using Krein space theory and Kolmogorov decomposition, obtaining a realization formula in a general setting encompassing the polydisc, the annulus, the Neil parabola, and certain quotient domains.

In the final part, we discuss various approximation results on quotients of the polydisc by certain imprimitive pseudo-reflection groups. Although the closed algebra generated by inner functions is not the whole of the algebra of bounded holomorphic functions, we present a Carathéodory-type approximation (approximation of contractive holomorphic functions by rational inner functions in the compact-open topology) and a Fisher-type approximation (approximation by convex combinations of rational inner functions). Further, we illustrate the structure of rational inner functions.

Then we focus on the Hankel operators on Hardy spaces associated with quotient domains obtained by the action of finite pseudo-reflection groups on bounded symmetric domains and characterize the boundedness and compactness of small Hankel operators via weak product spaces.

The failure of Nehari’s theorem for big Hankel operators on the Hardy space of the polydisc was proven by Cotlar-Sadosky, Bakonyi-Timotin, and Ahern-Youssfi. This phenomenon extends to quotients of the polydisc and the Euclidean ball, leading to non-projectivity of the Hardy modules in the category of Hilbert modules. Interestingly, the Carathéodory approximation helps us to identify a category of Hilbert modules (imposing some topological conditions) where projective objects exist.