The isometry degree of a computable copy of $\ell^p$

TL;DR

This paper investigates the minimal computational complexity needed to find isometries between computable copies of l^p, showing that for p  2, these degrees are exactly the c.e. degrees, and such degrees always exist.

Contribution

It establishes the existence of the isometry degree for computable copies of l^p and characterizes these degrees as c.e. degrees when p  2.

Findings

The isometry degree always exists for computable copies of l^p.

When p  2, the isometry degrees are exactly the c.e. degrees.

The paper provides a precise characterization of the computational complexity of isometries for l^p.

Abstract

When $p$ is a computable real so that $p \geq 1$, the isometry degree of a computable copy $\mathcal{B}$ of $\ell^p$ is defined to be the least powerful Turing degree that computes a linear isometry of $\ell^p$ onto $\mathcal{B}$. We show that this degree always exists and that when $p \neq 2$ these degrees are precisely the c.e. degrees.