Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 931–943 | Cite as

Canonical Decomposition of Operators Associated with the Symmetrized Polydisc

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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.

Keywords

Spectral set Symmetrized polydisc \(\Gamma _n\)-Contraction Canonical decomposition 

Mathematics Subject Classification

47A13 47A15 47A20 47A25 47A45 

Notes

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.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

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