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Dilation, functional model and a complete unitary invariant for C.0Γn-contractions.

Authors :
Pal, Sourav
Source :
Infinite Dimensional Analysis, Quantum Probability & Related Topics. Mar2023, Vol. 26 Issue 1, p1-20. 20p.
Publication Year :
2023

Abstract

A commuting tuple of operators (S 1 , ... , S n − 1 , P) , defined on a Hilbert space ℋ , for which the closed symmetrized polydisc Γ n = ∑ 1 ≤ i ≤ n z i , ∑ 1 ≤ i < j ≤ n z i z j , ... , ∏ i = 1 n z i : | z i | ≤ 1 , i = 1 , ... , n is a spectral set, is called a Γ n -contraction. A Γ n -contraction is said to be pure or C. 0 if P is C. 0 , that is, if P * n → 0 strongly as n → ∞. We show that for any Γ n -contraction (S 1 , ... , S n − 1 , P) , there is a unique operator tuple (A 1 , ... , A n − 1) that satisfies the operator identities S i − S n − i * P = D P A i D P , i = 1 , ... , n − 1. This unique tuple is called the fundamental operator tuple or ℱ O -tuple of (S 1 , ... , S n − 1 , P). With the help of the ℱ O -tuple, we construct an operator model for a C. 0 Γ n -contraction and show that there exist n − 1 operators C 1 , ... , C n − 1 such that each S i can be represented as S i = C i + P C n − i * . We find an explicit minimal dilation for a class of C. 0 Γ n -contractions whose ℱ O -tuples satisfy a certain condition. Also, we establish that the ℱ O -tuple of (S 1 * , ... , S n − 1 * , P *) together with the characteristic function of P constitutes a complete unitary invariant for the C. 0 Γ n -contractions. The entire program is an analog of the Sz.-Nagy–Foias theory for C. 0 contractions. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
02190257
Volume :
26
Issue :
1
Database :
Academic Search Index
Journal :
Infinite Dimensional Analysis, Quantum Probability & Related Topics
Publication Type :
Academic Journal
Accession number :
162642470
Full Text :
https://doi.org/10.1142/S0219025722500205