Rocky Mountain Journal of Mathematics

Dilation theory in finite dimensions: The possible, the impossible and the unknown

Eliahu Levy and Orr Moshe Shalit

Full-text: Open access

Abstract

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These results can be used to give very elementary proofs of sharpened versions of some von Neumann type inequalities, as well as some other striking consequences about polynomials and matrices. Exploring the limits of the finite dimensional approach sheds light on the difference between those techniques and phenomena in operator theory that are inherently infinite dimensional, and those that are not.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 1 (2014), 203-221.

Dates
First available in Project Euclid: 2 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1401740499

Digital Object Identifier
doi:10.1216/RMJ-2014-44-1-203

Mathematical Reviews number (MathSciNet)
MR3216017

Zentralblatt MATH identifier
1297.47011

Subjects
Primary: 47A20: Dilations, extensions, compressions
Secondary: 15A45: Miscellaneous inequalities involving matrices 47A57: Operator methods in interpolation, moment and extension problems [See also 30E05, 42A70, 42A82, 44A60]

Citation

Levy, Eliahu; Shalit, Orr Moshe. Dilation theory in finite dimensions: The possible, the impossible and the unknown. Rocky Mountain J. Math. 44 (2014), no. 1, 203--221. doi:10.1216/RMJ-2014-44-1-203. https://projecteuclid.org/euclid.rmjm/1401740499


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References

  • J. Agler and J.E. McCarthy, Distinguished varieties, Acta Math. {194 (2005), 133-153.
  • T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88-90.
  • W.B. Arveson, Dilation theory yesterday and today, in A glimpse of Hilbert space operators: Paul R. Halmos in memoriam, S. Axler, P. Rosenthal and D. Sarason, eds., Birkhäuser Verlag, 2010.
  • B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. Amer. Math. Soc. 348 (1996), 561-583.
  • B.V.R. Bhat and M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infinite Dimensional Analysis, Quantum Prob. Rel. Top. 3 (2000), 519-575.
  • M.D. Choi, Completely positive linear maps on complex matrices, Linear Alg. Appl. 10 (1975), 285-290.
  • M.D. Choi and C.K. Li, Constrained unitary dilations and numerical ranges, J. Oper. Theor. 46 (2001), 435-447.
  • D. Cichoń, J. Stochel and F. H. Szafraniec, Extending positive definiteness, Trans. Amer. Math. Soc. 363 (2011), 545-577.
  • S.W. Drury, Remarks on von Neumann's inequality, Lect. Notes Math. 995 (1983), Springer, Berlin.
  • E. Egerváry, On the contractive linear transformations of $n$-dimensional vector space, Acta Sci. Math. Szeg. 15 (1954), 178-182.
  • C. Foias and A.E. Frazho, The commutant lifting approach to interpolation problems, Oper. Theor. Adv. Appl. 44 (1990), Birkhäuser Verlag, Basel.
  • C. Foiaş and B. Sz.-Nagy, Harmonic analysis of operators in Hilbert space, North-Holland, Amsterdam, 1970.
  • P.R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 12-134.
  • –––, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933.
  • –––, A Hilbert space problem book, 2nd edition, Springer-Verlag, New York, 1982.
  • J.A. Holbrook, Inequalities of von Neumann type for small matrices, Lect. Notes Pure Appl. Math. 136 (1992), Dekker, New York.
  • –––, Schur norms and the multivariate von Neumann inequality, Oper. Theor. Adv. Appl. 127 (2001), Birkhäuser, Basel.
  • John E. McCarthy and Orr M. Shalit, Unitary $N$-dilations for tuples of commuting matrices, Proc. Amer. Math. Soc. 141 (2013), 563-571.
  • P. Muhly and B. Solel, Quantum Markov processes (Correspondences and dilations), Inter. J. Math. 13 (2002), 863-906.
  • D. Opěla, A generalization of Andô's theorem and Parrott's example, Proc. Amer. Math. Soc. 134 (2006), 2703-2710.
  • S. Parrott, Unitary dilations for commuting contractions, Pac. J. Math. 34 (1970), 481-490.
  • V.I. Paulsen, Completely bounded maps and operator algebras, Cambr. Stud. Adv. Math. 78 (2002), Cambridge University Press, Cambridge.
  • G. Pisier, Similarity problems and completely bounded maps, Lect. Notes Math. 1618 (1996), Springer-Verlag, Berlin.
  • M. Putinar, A note on Tchakaloff's theorem, Proc. Amer. Math. Soc. 125 (1997), 2409-2414.
  • D. SeLegue, Minimal dilations of CP maps and $C^*$-extension of the Szegö limit theorem, Ph.D dissertation, University of California, Berkeley, 1997.
  • O.M. Shalit, Representing a product system representation as a contractive semigroup and applications to regular isometric dilations, Canad. Math. Bull. 53 (2010), 550-563.
  • O.M. Shalit and B. Solel, Subproduct systems, Doc. Math. 14 (2009), 801-868.
  • J. Stochel and F.H. Szafraniec, Unitary dilation of several contractions, Oper. Theor. Adv. Appl. 127 (2001), 585-598, Birkhauser, Berlin.
  • B. Sz.-Nagy, Sur les contractions de l'espace de Hilbert, Acta Sci. Math. Szeg. 15 (1953), 87-92.
  • –––, Extensions of linear transformations in Hilbert space which extend beyond this space (Appendix to Functional analysis, Frigyes Riesz and Béla Sz.-Nagy), Frederick Ungar Publishing Co., New York, 1960.
  • V. Tchakaloff, Formules de cubature mécanique à coefficients non négatifs, Bull. Sci. Math. 81 (1957), 123-134.
  • N.Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Funct. Anal. 16 (1974), 83-100. \noindentstyle