Complete Norm Preserving Extensions of Holomorphic Functions

Jim Agler; Lukasz Kosinski; John E. McCarthy

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arXiv · 2204.08521

Complete Norm Preserving Extensions of Holomorphic Functions

Jim Agler·Lukasz Kosinski·John E. McCarthy

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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.

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Jim Agler, Lukasz Kosinski, John E. McCarthy. 2022-04-18. Complete Norm Preserving Extensions of Holomorphic Functions. https://arxiv.org/abs/2204.08521

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