## Von Neumann’s inequality for commuting operator-valued multishifts

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- by Rajeev Gupta, Surjit Kumar and Shailesh Trivedi PDF
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**147**(2019), 2599-2608 Request permission

## 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.## References

- Jim Agler and John E. McCarthy,
*Distinguished varieties*, Acta Math.**194**(2005), no. 2, 133–153. MR**2231339**, DOI 10.1007/BF02393219 - T. Andô,
*On a pair of commutative contractions*, Acta Sci. Math. (Szeged)**24**(1963), 88–90. MR**155193** - Franka Miriam Brückler,
*Tensor products of $C^*$-algebras, operator spaces and Hilbert $C^*$-modules*, Math. Commun.**4**(1999), no. 2, 257–268. MR**1746388** - Sameer Chavan, Deepak Kumar Pradhan, and Shailesh Trivedi,
*Multishifts on directed Cartesian products of rooted directed trees*, Dissertationes Math.**527**(2017), 102. MR**3740250**, DOI 10.4064/dm758-6-2017 - M. J. Crabb and A. M. Davie,
*von Neumann’s inequality for Hilbert space operators*, Bull. London Math. Soc.**7**(1975), 49–50. MR**365179**, DOI 10.1112/blms/7.1.49 - Raúl E. Curto,
*Applications of several complex variables to multiparameter spectral theory*, Surveys of some recent results in operator theory, Vol. II, Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 25–90. MR**976843** - Raúl E. Curto and Norberto Salinas,
*Generalized Bergman kernels and the Cowen-Douglas theory*, Amer. J. Math.**106**(1984), no. 2, 447–488. MR**737780**, DOI 10.2307/2374310 - S. W. Drury,
*Remarks on von Neumann’s inequality*, Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981) Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 14–32. MR**717226**, DOI 10.1007/BFb0061886 - R. Gupta,
*The Carathéodory-Fejér interpolation problems and the von-Neumann inequality*, thesis, 2015, arXiv:1508.07199, 2015. - Rajeev Gupta,
*An improvement on the bound for $C_2(n)$*, Acta Sci. Math. (Szeged)**83**(2017), no. 1-2, 263–269. MR**3701044**, DOI 10.14232/actasm-015-088-4 - R. Gupta and S. Ray,
*On a question of N. Th. Varopoulos and the constant $C_2(n)$*, Ann. Inst. Fourier (Grenoble),**68**(2018), no. 6, 2613–2634, DOI 10.5802/aif.3218. - Greg Knese,
*The von Neumann inequality for $3\times 3$ matrices*, Bull. Lond. Math. Soc.**48**(2016), no. 1, 53–57. MR**3455747**, DOI 10.1112/blms/bdv087 - Michael Hartz,
*Von Neumann’s inequality for commuting weighted shifts*, Indiana Univ. Math. J.**66**(2017), no. 4, 1065–1079. MR**3689326**, DOI 10.1512/iumj.2017.66.6074 - John A. Holbrook,
*Schur norms and the multivariate von Neumann inequality*, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 375–386. MR**1902811** - Zenon Jan Jabłoński,
*Hyperexpansive operator valued unilateral weighted shifts*, Glasg. Math. J.**46**(2004), no. 2, 405–416. MR**2062623**, DOI 10.1017/S0017089504001892 - Nicholas P. Jewell and A. R. Lubin,
*Commuting weighted shifts and analytic function theory in several variables*, J. Operator Theory**1**(1979), no. 2, 207–223. MR**532875** - Alan Lambert,
*Unitary equivalence and reducibility of invertibly weighted shifts*, Bull. Austral. Math. Soc.**5**(1971), 157–173. MR**295128**, DOI 10.1017/S000497270004702X - A. L. Lambert and T. R. Turner,
*The double commutant of invertibly weighted shifts*, Duke Math. J.**39**(1972), 385–389. MR**310683** - John E. McCarthy and Orr Moshe Shalit,
*Unitary $N$-dilations for tuples of commuting matrices*, Proc. Amer. Math. Soc.**141**(2013), no. 2, 563–571. MR**2996961**, DOI 10.1090/S0002-9939-2012-11714-9 - Vern Paulsen,
*Completely bounded maps and operator algebras*, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR**1976867** - Marek Ptak,
*On the reflexivity of multigenerator algebras*, Dissertationes Math. (Rozprawy Mat.)**378**(1998), 61. MR**1655865** - B. Krishna Das, Jaydeb Sarkar, and Srijan Sarkar,
*Factorizations of contractions*, Adv. Math.**322**(2017), 186–200. MR**3720797**, DOI 10.1016/j.aim.2017.10.010 - Allen L. Shields,
*Weighted shift operators and analytic function theory*, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR**0361899** - Béla Sz.-Nagy,
*Sur les contractions de l’espace de Hilbert*, Acta Sci. Math. (Szeged)**15**(1953), 87–92 (French). MR**58128** - Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy,
*Harmonic analysis of operators on Hilbert space*, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR**2760647**, DOI 10.1007/978-1-4419-6094-8 - N. Th. Varopoulos,
*On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory*, J. Functional Analysis**16**(1974), 83–100. MR**0355642**, DOI 10.1016/0022-1236(74)90071-8 - N. Th. Varopoulos,
*On a commuting family of contractions on a Hilbert space*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 9, 1283–1285. MR**430824** - Johann von Neumann,
*Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes*, Math. Nachr.**4**(1951), 258–281 (German). MR**43386**, DOI 10.1002/mana.3210040124

## Additional Information

**Rajeev Gupta**- Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016 India
- MR Author ID: 1231493
- 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
- MR Author ID: 1064875
- Email: shailtr@iitk.ac.in
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 2599-2608 - MSC (2010): Primary 47B37; Secondary 47A13
- DOI: https://doi.org/10.1090/proc/14410
- MathSciNet review: 3951435