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Von Neumann's inequality for commuting operator-valued multishifts


Authors: Rajeev Gupta, Surjit Kumar and Shailesh Trivedi
Journal: Proc. Amer. Math. Soc. 147 (2019), 2599-2608
MSC (2010): Primary 47B37; Secondary 47A13
DOI: https://doi.org/10.1090/proc/14410
Published electronically: February 20, 2019
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Abstract: Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In fact, we show that if $ A$ and $ B$ are commuting contractive $ d$-tuples of operators such that $ B$ satisfies the matrix-version of von Neumann's inequality and $ (1, \ldots , 1)$ is in the algebraic spectrum of $ B$, then the tensor product $ A \otimes B$ satisfies von Neumann's inequality if and only if $ A$ satisfies von Neumann's inequality. We also exhibit several families of operator-valued multishifts for which von Neumann's inequality always holds.


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Additional Information

Rajeev Gupta
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016 India
Email: rajeevg@iitk.ac.in

Surjit Kumar
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore, 560012 India
Email: surjitkumar@iisc.ac.in

Shailesh Trivedi
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016 India
Email: shailtr@iitk.ac.in

DOI: https://doi.org/10.1090/proc/14410
Keywords: Operator-valued multishift, von Neumann's inequality, tensor product
Received by editor(s): June 13, 2018
Received by editor(s) in revised form: September 25, 2018
Published electronically: February 20, 2019
Additional Notes: The work of the first and second authors was supported by Inspire Faculty Fellowship (Ref. No. DST/INSPIRE/04/2017/002367, DST/INSPIRE/04/2016/001008)
The third author was supported by the National Post-doctoral Fellowship (Ref. No. PDF/2016/001681), SERB
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society