1. A Nagy–Foias Program for a C.N.U. Γn-Contraction.
- Author
-
Bisai, Bappa and Pal, Sourav
- Abstract
A tuple of commuting Hilbert space operators (S 1 , … , S n - 1 , P) having the closed symmetrized polydisc Γ n = ∑ i = 1 n z i , ∑ 1 ≤ i < j ≤ n z i z j , … , ∏ i = 1 n z i : | z i | ≤ 1 , 1 ≤ i ≤ n - 1
as a spectral set is called a Γ n -contraction. From the literature we have that a point (s 1 , … , s n - 1 , p) in Γ n can be represented as s i = c i + p c n - i for some (c 1 , … , c n - 1 ) ∈ Γ n - 1 . We construct a minimal Γ n -isometric dilation for a particular class of c.n.u. Γ n -contractions (S 1 , … , S n - 1 , P) and obtain a functional model for them. With the help of this model we express each S i as S i = C i + P C n - i , which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. Γ n -contractions satisfying S i ∗ P = P S i ∗ for each i. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that S i ∗ P = P S i ∗ . We apply this abstract model to achieve a complete unitary invariant for such c.n.u. Γ n -contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple (S 1 , … , S n - 1 , P) becomes a Γ n -contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF