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Operators with analytic orbit for the torus action

Volume 243 / 2018

Rodrigo A. H. M. Cabral, Severino T. Melo Studia Mathematica 243 (2018), 243-250 MSC: Primary 47G30; Secondary 35S05, 58G15. DOI: 10.4064/sm8767-10-2017 Published online: 11 May 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.

Authors

  • Rodrigo A. H. M. CabralInstituto de Matemática e Estatística
    Universidade de São Paulo
    Rua do Matão 1010
    São Paulo, SP 05508-090, Brazil
    e-mail
  • Severino T. MeloInstituto de Matemática e Estatística
    Universidade de São Paulo
    Rua do Matão 1010
    São Paulo, SP 05508-090, Brazil
    e-mail

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