Spectral and mapping properties of the inverse Ornstein--Uhlenbeck operator

Abstract

We investigate the spectral and boundedness properties of the inverse Ornstein--Uhlenbeck operator 𝒜^Q,B and its associated semigroup in L^p spaces with respect to the inverse Gaussian measure. We establish strong continuity and contractivity of this semigroup in L^p for p ∈ [1, ∞), and obtain a complete characterization of the spectrum of 𝒜^Q,B. For p ∈ (1, ∞), the spectrum is independent of p, consists entirely of eigenvalues and exhibits a spectral gap. For p=1, the spectrum is instead the left half-plane. Furthermore, we discuss ultracontractivity and hypercontractivity properties of the semigroup.

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BibTeX

@misc{bruno2026spectral,
  title = {{Spectral and mapping properties of the inverse Ornstein--Uhlenbeck operator}},
  author = {Tommaso Bruno and Valentina Casarino and Paolo Ciatti and Peter Sjögren},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06826},
  eprint = {2611.06826},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06826},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Spectral and mapping properties of the inverse Ornstein--Uhlenbeck operator
AU  - Tommaso Bruno
AU  - Valentina Casarino
AU  - Paolo Ciatti
AU  - Peter Sjögren
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06826
UR  - https://arxiv.org/abs/2611.06826
AB  - We investigate the spectral and boundedness properties of the inverse Ornstein--Uhlenbeck operator 𝒜^Q,B and its associated semigroup in L^p spaces with respect to the inverse Gaussian measure. We establish strong continuity and contractivity of this semigroup in L^p for p ∈ [1, ∞), and obtain a complete characterization of the spectrum of 𝒜^Q,B. For p ∈ (1, ∞), the spectrum is independent of p, consists entirely of eigenvalues and exhibits a spectral gap. For p=1, the spectrum is instead the left half-plane. Furthermore, we discuss ultracontractivity and hypercontractivity properties of the semigroup.
ER  -

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