Operator theory and 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 $|{\delta }^l(z)|<1,1\le l\le m$, and prove a representation formula for it. We give conditions whereby every function that is analytic on a neighborhood of $|{\delta }^l(z)|\le 1,1\le l\le m$ 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.