## Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

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- by Sibaprasad Barik, B. Krishna Das, Kalpesh J. Haria and Jaydeb Sarkar PDF
- Trans. Amer. Math. Soc.
**372**(2019), 1429-1450 Request permission

## Abstract:

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb {C}[z]$ or $\mathbb {C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of isometric dilations, von Neumann inequality, and a refined version of von Neumann inequality for a large class of $n$-tuples, $n \geq 3$, of commuting contractions.## References

- Jim Agler and John E. McCarthy,
*Nevanlinna-Pick interpolation on the bidisk*, J. Reine Angew. Math.**506**(1999), 191–204. MR**1665697**, DOI 10.1515/crll.1999.004 - Jim Agler and John E. McCarthy,
*Pick interpolation and Hilbert function spaces*, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR**1882259**, DOI 10.1090/gsm/044 - 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** - William B. Arveson,
*Subalgebras of $C^{\ast }$-algebras*, Acta Math.**123**(1969), 141–224. MR**253059**, DOI 10.1007/BF02392388 - William Arveson,
*Subalgebras of $C^{\ast }$-algebras. II*, Acta Math.**128**(1972), no. 3-4, 271–308. MR**394232**, DOI 10.1007/BF02392166 - J. A. Ball, W. S. Li, D. Timotin, and T. T. Trent,
*A commutant lifting theorem on the polydisc*, Indiana Univ. Math. J.**48**(1999), no. 2, 653–675. MR**1722812**, DOI 10.1512/iumj.1999.48.1708 - C. A. Berger, L. A. Coburn, and A. Lebow,
*Representation and index theory for $C^*$-algebras generated by commuting isometries*, J. Functional Analysis**27**(1978), no. 1, 51–99. MR**0467392**, DOI 10.1016/0022-1236(78)90019-8 - Man-Duen Choi and Kenneth R. Davidson,
*A $3\times 3$ dilation counterexample*, Bull. Lond. Math. Soc.**45**(2013), no. 3, 511–519. MR**3065020**, DOI 10.1112/blms/bds109 - Mischa Cotlar and Cora Sadosky,
*Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy-Foias theorem for two pairs of dilations*, J. Funct. Anal.**111**(1993), no. 2, 473–488. MR**1203463**, DOI 10.1006/jfan.1993.1022 - 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 - R. E. Curto and F.-H. Vasilescu,
*Standard operator models in the polydisc*, Indiana Univ. Math. J.**42**(1993), no. 3, 791–810. MR**1254118**, DOI 10.1512/iumj.1993.42.42035 - B. Krishna Das and Jaydeb Sarkar,
*Ando dilations, von Neumann inequality, and distinguished varieties*, J. Funct. Anal.**272**(2017), no. 5, 2114–2131. MR**3596718**, DOI 10.1016/j.jfa.2016.08.008 - 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 - Louis de Branges and James Rovnyak,
*Square summable power series*, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966. MR**0215065** - 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 - Jörg Eschmeier and Mihai Putinar,
*Spherical contractions and interpolation problems on the unit ball*, J. Reine Angew. Math.**542**(2002), 219–236. MR**1880832**, DOI 10.1515/crll.2002.007 - Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov, and Hugo J. Woerdeman,
*Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality*, J. Funct. Anal.**256**(2009), no. 9, 3035–3054. MR**2502431**, DOI 10.1016/j.jfa.2008.09.012 - John A. Holbrook,
*Inequalities of von Neumann type for small matrices*, Function spaces (Edwardsville, IL, 1990) Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992, pp. 189–193. MR**1152347** - 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** - 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 - Łukasz Kosiński,
*Three-point Nevanlinna-Pick problem in the polydisc*, Proc. Lond. Math. Soc. (3)**111**(2015), no. 4, 887–910. MR**3407188**, DOI 10.1112/plms/pdv045 - V. Müller and F.-H. Vasilescu,
*Standard models for some commuting multioperators*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 979–989. MR**1112498**, DOI 10.1090/S0002-9939-1993-1112498-0 - Béla Sz.-Nagy and Ciprian Foiaş,
*Harmonic analysis of operators on Hilbert space*, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR**0275190** - Stephen Parrott,
*Unitary dilations for commuting contractions*, Pacific J. Math.**34**(1970), 481–490. MR**268710** - Gilles Pisier,
*Similarity problems and completely bounded maps*, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 1996. MR**1441076**, DOI 10.1007/978-3-662-21537-1 - J. Sarkar,
*An Introduction to Hilbert module approach to multivariable operator theory*, Operator Theory, Springer Basel (2015), 969–1033. - 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

## Additional Information

**Sibaprasad Barik**- Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
- Email: sibaprasadbarik00@gmail.com
**B. Krishna Das**- Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
- MR Author ID: 999492
- Email: dasb@math.iitb.ac.in, bata436@gmail.com
**Kalpesh J. Haria**- Affiliation: School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, 175005, Himachal Pradesh, India
- MR Author ID: 1037705
- ORCID: setImmediate$0.43056093711685606$1
- Email: kalpesh@iitmandi.ac.in, hikalpesh.haria@gmail.com
**Jaydeb Sarkar**- Affiliation: Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
- MR Author ID: 773222
- Email: jay@isibang.ac.in, jaydeb@gmail.com
- Received by editor(s): November 1, 2017
- Received by editor(s) in revised form: June 18, 2018
- Published electronically: April 4, 2019
- Additional Notes: The research of the first author is supported by Council of Scientific & Industrial Research (CSIR) Fellowship

The research of the second author is supported by DST-INSPIRE Faculty Fellowship No. DST/INSPIRE/04/2015/001094

The research work of the third author is supported by DST-INSPIRE Faculty Fellowship No. DST/INSPIRE/04/2014/002624

The research of the fourth author is supported in part by Mathematical Research Impact Centric Support (MATRICS) grant, File No : MTR/2017/000522, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India, and NBHM (National Board of Higher Mathematics, India) Research Grant NBHM/R.P.64/2014 - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 1429-1450 - MSC (2010): Primary 47A13, 47A20, 47A45, 47A56, 46E22, 47B32, 32A35, 32A70
- DOI: https://doi.org/10.1090/tran/7676
- MathSciNet review: 3968807