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Domination of semigroups on standard forms of von Neumann algebras

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Abstract

Consider \((T_t)_{t\ge 0}\) and \((S_t)_{t\ge 0}\) as real \(C_0\)-semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup \((T_t)_{t\ge 0}\) by \((S_t)_{t\ge 0}\), which means that \(-S_t v\le T_t u\le S_t v\) holds for all \(t\ge 0\) and all real u and v that satisfy \(-v\le u\le v\). This characterisation extends the Ouhabaz characterisation for semigroup domination to the non-commutative \(L^2\)-spaces. Additionally, we present a simpler characterisation when both semigroups are positive as well as consider the setting in which \((T_t)_{t\ge 0}\) need not be real.

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Acknowledgements

The first author is grateful to Melchior Wirth for various fruitful discussions about standard forms and for pointing out the reference [5].

Funding

The funding has been received from SERB, DST, India with Grant no. VAJRA - VJR/2018/000127.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Twente, 217, 7500, AE, Enschede, The Netherlands

    Sahiba Arora

  2. Technische Universität Dresden Institut für Analysis Fakultät für Mathematik, 01062, Dresden, Germany

    Ralph Chill

  3. Department of Mathematics, University of Delhi, South Campus, New Delhi, 110021, India

    Sachi Srivastava

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  1. Sahiba Arora
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  2. Ralph Chill
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  3. Sachi Srivastava
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Correspondence to Sachi Srivastava.

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This project is supported by the VAJRA scheme VJR/2018/000127, of the Science and Engineering Research Board, Department of Science and Technology, Govt. of India. The second and third authors gratefully acknowledge the support from VAJRA.

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Arora, S., Chill, R. & Srivastava, S. Domination of semigroups on standard forms of von Neumann algebras. Arch. Math. 121, 715–729 (2023). https://doi.org/10.1007/s00013-023-01946-y

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  • DOI: https://doi.org/10.1007/s00013-023-01946-y

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