Abstract
Let \(\mathcal {M}\) be a von Neumann algebra with a normal faithful semifinite trace. In this paper, we consider that in n-tuples of noncommutative \(L_p\)-spaces \(l_s^{(n)}(L_p(\mathcal {M}))\), the norm is invariant under the action of invertible elements in \(\mathcal {M}\). Then we prove that the complex interpolating theorem in the case of \(l_s^{(n)}(L_p(\mathcal {M}))\). Using this result, we obtain that Clarkson’s inequalities for n-tuples of operators with weighted norm of noncommutative \(L_p\)-spaces, where the weight being a positive invertible operator in \(\mathcal {M}\).
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Acknowledgements
This work is supported by “Research Project Supported by Shanxi Scholarship Council of China” no. 2020-031, “Fundamental Research Program of Shanxi Province” no. 202103021224104 and 202203021212211, “Chunhui Plan of Ministry of Education of China” no. 202200101 and “Scientific Research Fund of Taiyuan University of Technology” no. 2022QN098. The author would like to thank Professor Yazhou Han for the discussion of this paper.
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Zhang, F. Geometric Interpolation in n-Tuples of Noncommutative \(L_p\)-Spaces. Complex Anal. Oper. Theory 18, 89 (2024). https://doi.org/10.1007/s11785-024-01535-z
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DOI: https://doi.org/10.1007/s11785-024-01535-z