Operators with analytic orbit for the torus action
Volume 243 / 2018
Abstract
The class of bounded operators on $L^2({\mathbb T}^{n})$ which have an analytic orbit under the action of ${\mathbb T}^{n}$ by conjugation with the translation operators is shown to coincide with the class of zero-order pseudodifferential operators whose discrete symbol $(a_j)_{j\in {\mathbb Z}^n}$ is uniformly analytic, in the sense that there exists $C \gt 1$ such that the derivatives of $a_j$ satisfy $|\partial ^\alpha a_j(x)|\leq C^{1+|\alpha |}\alpha !$ for all $x\in {\mathbb T}^{n}$, all $j\in {\mathbb Z}^n$ and all $\alpha \in {\mathbb N}^n$. It then follows that this class of analytic pseudodifferential operators is a spectrally invariant $^{*}$-subalgebra of the algebra of bounded operators on $L^2({\mathbb T}^{n})$, dense (in norm topology) in the algebra of $\rho =\delta =0$ Hörmander-type operators.
