Complete Norm Preserving Extensions of Holomorphic Functions

TL;DR

The paper establishes conditions for extending bounded holomorphic functions from analytic subvarieties to larger pseudoconvex domains while preserving norms, with specific results for intersecting disks and the symmetrized bidisk.

Contribution

It provides a comprehensive method for norm-preserving extensions of holomorphic functions from subvarieties to pseudoconvex sets, including new insights for matrix-valued functions and special domains.

Findings

Existence of pseudoconvex sets allowing isometric extension of bounded matrix-valued holomorphic functions.

Extension property holds for intersecting disks under certain conditions.

Linear isometric extension fails for the symmetrized bidisk when functions vanish at a point.

Abstract

We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $\Omega$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $\Omega$. We prove that if $V$ is two analytic disks intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to $\Omega$, then so does every matrix-valued function. In the special case that $\Omega$ is the symmetrized bidisk, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.