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Infinite order psi DOs : composition with entire functions, new Shubin-Sobolev spaces, and index theorem

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Abstract
We study global regularity and spectral properties of power series of the Weyl quantisation a(w), where a(x, xi) is a classical elliptic Shubin polynomial. For a suitable entire function P, we associate two natural infinite order operators to a(w), P(a(w)) and (P. a)(w), and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to infinity 8 for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of f-Gamma(Ap,rho)*(infinity)-elliptic symbols, where f is a function of ultrapolynomial growth and Gamma(Ap,rho)*(infinity) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hormander integral formula.
Keywords
Infinite order pseudo-differential operators, Shubin type operators, power series of operators, Gelfand-Shilov regularity, Shubin-Sobolev spaces, index theorems, PSEUDODIFFERENTIAL-OPERATORS, ULTRADISTRIBUTIONS, ASYMPTOTICS, BOLTZMANN

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MLA
Pilipović, Stevan, et al. “Infinite Order Psi DOs : Composition with Entire Functions, New Shubin-Sobolev Spaces, and Index Theorem.” ANALYSIS AND MATHEMATICAL PHYSICS, vol. 11, no. 3, 2021, doi:10.1007/s13324-021-00545-w.
APA
Pilipović, S., Prangoski, B., & Vindas Diaz, J. (2021). Infinite order psi DOs : composition with entire functions, new Shubin-Sobolev spaces, and index theorem. ANALYSIS AND MATHEMATICAL PHYSICS, 11(3). https://doi.org/10.1007/s13324-021-00545-w
Chicago author-date
Pilipović, Stevan, Bojan Prangoski, and Jasson Vindas Diaz. 2021. “Infinite Order Psi DOs : Composition with Entire Functions, New Shubin-Sobolev Spaces, and Index Theorem.” ANALYSIS AND MATHEMATICAL PHYSICS 11 (3). https://doi.org/10.1007/s13324-021-00545-w.
Chicago author-date (all authors)
Pilipović, Stevan, Bojan Prangoski, and Jasson Vindas Diaz. 2021. “Infinite Order Psi DOs : Composition with Entire Functions, New Shubin-Sobolev Spaces, and Index Theorem.” ANALYSIS AND MATHEMATICAL PHYSICS 11 (3). doi:10.1007/s13324-021-00545-w.
Vancouver
1.
Pilipović S, Prangoski B, Vindas Diaz J. Infinite order psi DOs : composition with entire functions, new Shubin-Sobolev spaces, and index theorem. ANALYSIS AND MATHEMATICAL PHYSICS. 2021;11(3).
IEEE
[1]
S. Pilipović, B. Prangoski, and J. Vindas Diaz, “Infinite order psi DOs : composition with entire functions, new Shubin-Sobolev spaces, and index theorem,” ANALYSIS AND MATHEMATICAL PHYSICS, vol. 11, no. 3, 2021.
@article{8737323,
  abstract     = {{We study global regularity and spectral properties of power series of the Weyl quantisation a(w), where a(x, xi) is a classical elliptic Shubin polynomial. For a suitable entire function P, we associate two natural infinite order operators to a(w), P(a(w)) and (P. a)(w), and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to infinity 8 for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of f-Gamma(Ap,rho)*(infinity)-elliptic symbols, where f is a function of ultrapolynomial growth and Gamma(Ap,rho)*(infinity) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hormander integral formula.}},
  articleno    = {{109}},
  author       = {{Pilipović, Stevan and Prangoski, Bojan and Vindas Diaz, Jasson}},
  issn         = {{1664-2368}},
  journal      = {{ANALYSIS AND MATHEMATICAL PHYSICS}},
  keywords     = {{Infinite order pseudo-differential operators,Shubin type operators,power series of operators,Gelfand-Shilov regularity,Shubin-Sobolev spaces,index theorems,PSEUDODIFFERENTIAL-OPERATORS,ULTRADISTRIBUTIONS,ASYMPTOTICS,BOLTZMANN}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{48}},
  title        = {{Infinite order psi DOs : composition with entire functions, new Shubin-Sobolev spaces, and index theorem}},
  url          = {{http://doi.org/10.1007/s13324-021-00545-w}},
  volume       = {{11}},
  year         = {{2021}},
}

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