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