Differential operators on Hermite Sobolev spaces

@article{Bhar2015DifferentialOO,
  title={Differential operators on Hermite Sobolev spaces},
  author={Suprio Bhar and Bhaskaran Rajeev},
  journal={Proceedings - Mathematical Sciences},
  year={2015},
  volume={125},
  pages={113-125}
}
In this paper, we compute the Hilbert space adjoint ∂∗ of the derivative operator ∂ on the Hermite Sobolev spaces Sq$\mathcal {S}_{q}$. We use this calculation to give a different proof of the ‘monotonicity inequality’ for a class of differential operators (L,A) for which the inequality was proved in Infin. Dimens. Anal. Quantum Probab. Relat. Top.2(4) (2009) 515–591. We also prove the monotonicity inequality for (L,A), when these correspond to the Ornstein–Uhlenbeck diffusion. 

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