Admissible Fundamental Operators and Models for Γ_{E(3; 3; 1, 1, 1)}-contraction and Γ_{E(3; 2; 1, 2)}-contraction

2025/11/04 by Pal, Avijit, Paul, Bhaskar
Mathematics · #Holomorphic and Operator Theory #Advanced Operator Algebra Research #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2511.02635

Abstract

We show that for a given pure contraction T₇ acting on a Hilbert space H, if (F₁, …, F₆) ∈ B(D_T^*7) with [F_i, F_j] = 0, [F^*_i, F_7-j] = [F^*_j, F_7-i],w(F^*_i + F_7-iz) \leqslant 1 and these operators satisfy (F^*_i + F_7-iz)Θ_T7(z) = Θ_T7(z)(F_i + F^*_7-iz) for all z ∈ \mathbbD for 1 \leqslant i, j \leqslant 6 for some (F₁, …, F₆) ∈ B(D_T7) with w(F^*_i + F_7-iz) \leqslant 1 for 1 \leqslant i \leqslant 6, then there exists a Γ_{E(3; 3; 1, 1, 1)}-contraction (T₁, …, T₇) such that F₁, …, F₆ are the fundamental operators of (T₁, …, T₇) and F₁, …, F₆ are the fundamental operators of (T^*₁, …, T^*₇). We also prove similar type of result for pure Γ_{E(3; 2; 1, 2)}-contraction. We explicitly construct a Γ_{E(3; 3; 1, 1, 1)}-unitary (respectively, a Γ_{E(3; 2; 1, 2)}-unitary) starting from a Γ_{E(3; 3; 1, 1, 1)}-contraction (respectively, a Γ_{E(3; 2; 1, 2)}-contraction). Further, we develop functional models for general Γ_{E(3; 3; 1, 1, 1)}-isometries (respectively, Γ_{E(3; 2; 1, 2)}-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for Γ_{E(3; 3; 1, 1, 1)}-contractions (respectively, Γ_{E(3; 2; 1, 2)}-contractions). Finally, we present a Schaffer-type model for the Γ_{E(3; 3; 1, 1, 1)}-isometric dilation (respectively, the Γ_{E(3; 2; 1, 2)}-isometric dilation).

Citations

• Dilation theory and functional models for tetrablock contractions
• Rational dilation of tetrablock contractions revisited
• Functional Models and Invariant Subspaces for Pairs of Commuting Contractions
• Contractivity and complete contractivity for finite dimensional Banach\n Spaces
• Canonical decomposition of a tetrablock contraction and operator model
• Operator Theory on Symmetrized Bidisc
• Geometry of the symmetrized polydisc
• The failure of rational dilation on a triply connected domain

Related

• Functional Models for Γ_{E(3; 3; 1, 1, 1)}-contraction, Γ_{E(3; 2; 1, 2)}-contraction and Tetrablock contraction
• Canonical Decompositions and Conditional Dilations of Γ_{E(3; 3; 1, 1, 1)}-Contraction and Γ_{E(3; 2; 1, 2)}-Contraction
• Necessary Conditions for Γ_{E(3; 3; 1, 1, 1)}-Isometric Dilation, Γ_{E(3; 2; 1, 2)}-Isometric Dilation and \mathcalP-Isometric Dilation
• Operators on Hilbert Space having Γ_{E(3; 3; 1, 1, 1)} and Γ_{E(3; 2; 1, 2)} as Spectral Sets
• Necessary conditions for existence of Γ_n-contractions and examples of Γ₃-contractions