Operators with analytic orbit for the torus action
  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 $L^2(T^n)$. 
Rodrigo A. H. M. Cabral, Severino T. Melo
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