Operator theory and the Oka extension theorem

TL;DR

This paper develops an algebraic framework for analytic functions on polyhedra, proves a representation formula, and provides a bounded Oka extension theorem with an operator calculus.

Contribution

It introduces a new algebra $ ext{H}^ ext{d}_ ext{g}$, proves a representation formula, and extends the Oka theorem with bounds using this algebra.

Findings

Established a representation formula for $ ext{H}^ ext{d}_ ext{g}$.

Proved conditions for analytic functions to belong to $ ext{H}^ ext{d}_ ext{g}$.

Provided a bounded version of the Oka extension theorem.

Abstract

For $\delta$ an $m$-tuple of analytic functions, we define an algebra $\hidg$, contained in the bounded analytic functions on the analytic polyhedron $ {|\delta^l(z)| < 1, \ 1 \leq l \leq m}$, and prove a representation formula for it. We give conditions whereby every function that is analytic on a neighborhood of $ {|\delta^l(z)| \leq 1, \ 1 \leq l \leq m}$ is actually in $\hidg$. We use this to give a proof of the Oka extension theorem with bounds. We define an $\hidg$ functional calculus for operators.