Operators with analytic orbit for the torus action

2016/10/19 by Cabral, Rodrigo A. H. M., Melo, Severino T.
#35S05 #47G30 #58G15 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1610.06439

Abstract

Let Tⁿ denote the n-dimensional torus. The class of the bounded operators on L²(Tⁿ) with analytic orbit under the action of Tⁿ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on Tⁿ whose discrete symbol (a_j)_{j∈ Zⁿ} is uniformly analytic, in the sense that there exists C>1 such that the derivatives of a_j satisfy |∂^αa_j(x)|≤ C^1+|α|α! for all x∈ Tⁿ, all j∈ Zⁿ and all α∈ Nⁿ. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on L²(Tⁿ).

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