Operators with analytic orbit for the torus action
Tom 243 / 2018
Streszczenie
The class of bounded operators on 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.
