Operators with analytic orbit for the torus action

TL;DR

This paper characterizes operators with analytic orbits under torus action as a class of zero-order pseudodifferential operators with uniformly analytic symbols, showing they form a spectrally invariant subalgebra.

Contribution

It establishes an equivalence between operators with analytic orbits and a specific class of pseudodifferential operators with analytic symbols, extending understanding of their algebraic properties.

Findings

Operators with analytic orbit are zero-order pseudodifferential operators.

These operators have uniformly analytic discrete symbols.

The class forms a spectrally invariant *-subalgebra.

Abstract

Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\partial^\alpha a_j(x)|\leq C^{1+|\alpha|}\alpha!$ for all $x\in T^n$, all $j\in Z^n$ and all $\alpha\in N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.