Abstract
A tuple of commuting operators \((S_1,\dots ,S_{n-1},P)\) for which the closed symmetrized polydisc \(\Gamma _n\) is a spectral set is called a \(\Gamma _n\)-contraction. We show that every \(\Gamma _n\)-contraction admits a decomposition into a \(\Gamma _n\)-unitary and a completely non-unitary \(\Gamma _n\)-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set \(\Gamma _n\) and \(\Gamma _n\)-contractions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
08 July 2023
A Correction to this paper has been published: https://doi.org/10.1007/s11785-023-01349-5
References
Agler, J., Lykova, Z.A., Young, N.J.: A case of \(\mu \)-synthesis as a quadratic semidefinite program. SIAM J. Control Optim. 51(3), 2472–2508 (2013)
Agler, J., Young, N.J.: A commutant lifting theorem for a domain in \({\mathbb{C}}^2\) and spectral interpolation. J. Funct. Anal. 161, 452–477 (1999)
Agler, J., Young, N.J.: A model theory for \(\Gamma \)-contractions. J. Oper. Theory 49, 45–60 (2003)
Bhatia, R.: Positive Definite Matrices, Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2007)
Bhattacharyya, T., Pal, S., Roy, S.S.: Dilations of \(\Gamma \)-contractions by solving operator equations. Adv. Math. 230, 577–606 (2012)
Bhattacharyya, T., Pal, S.: A functional model for pure \(\Gamma \)-contractions. J. Oper. Theory 71, 327–339 (2014)
Biswas, S., Shyam Roy, S.: Functional models for \(\Gamma _n\)-contractions and characterization of \(\Gamma _n\)-isometries. J. Funct. Anal 266, 6224–6255 (2014)
Bercovici, H., Foias, C., Kerchy, L., Nagy, S.: Harmonic Analysis of Operators on Hilbert Space, Universitext. Springer, New York (2010)
Costara, C.: On the spectral Nevanlinna–Pick problem. Stud. Math. 170, 23–55 (2005)
Gamelin, T.: Uniform Algebras. Prentice Hall, Upper Saddle River (1969)
Pal, S.: From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator models. N. Y. J. Math. 20, 545–564 (2014)
Pal, S.: On decomposition of operators having \(\Gamma _3\) as a spectral set. Oper. Matrices 11, 891–899 (2017)
Pal, S., Shalit, O.M.: Spectral sets and distinguished varieties in the symmetrized bidisc. J. Funct. Anal. 266, 5779–5800 (2014)
Vasilescu, F.H.: Analytic Functional Calculus and Spectral Decompositions. Editura Academiei: Bucuresti, D. Reidel Publishing Company, Romania (1982)
von Neumann, J.: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951)
Acknowledgements
The author is thankful to the referee for making numerous invaluable comments on the article. The referee’s suggestions helped in refining the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Joseph Ball.
The author is supported by Seed Grant of IIT Bombay, CPDA and the INSPIRE Faculty Award (Award No. DST/INSPIRE/04/2014/001462) of DST, India.
Rights and permissions
About this article
Cite this article
Pal, S. Canonical Decomposition of Operators Associated with the Symmetrized Polydisc. Complex Anal. Oper. Theory 12, 931–943 (2018). https://doi.org/10.1007/s11785-017-0721-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-017-0721-1