Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set

Autor: Gadadhar Misra, Avijit Pal
Rok vydání: 2018
Předmět:
Zdroj: Journal d'Analyse Mathématique. 136:31-54
ISSN: 1565-8538
0021-7670
DOI: 10.1007/s11854-018-0054-7
Popis: For any bounded domain Ω in ℂm, let B1(Ω) denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple T in the Cowen-Douglas class B1(Ω), let NT (w) denote the restriction of T to the subspace $$\cap^m_{i,j=1}{\rm{ker}}(T_i-w_iI)(T_j-w_jI)$$ . This commuting m-tuple NT (w) of m + 1 dimensional operators induces a homomorphism $${\rho _{{N_T}\left( w \right)}}$$ of the polynomial ring P[z1, · · ·, zm], namely, $${\rho _{{N_T}\left( w \right)}}$$ (p) = p(NT (w)), p ∈ P[z1, · · ·, zm]. We study the contractivity and complete contractivity of the homomorphism $${\rho _{{N_T}\left( w \right)}}$$ . Starting from the homomorphism $${\rho _{{N_T}\left( w \right)}}$$ , we construct a natural class of homomorphisms $$\rho_{N^{(\lambda)}(w)}$$ , λ > 0, and relate the properties of $$\rho_{N^{(\lambda)}(w)}$$ to those of $${\rho _{{N_T}\left( w \right)}}$$ . Explicit examples arising from the multiplication operators on the Bergman space of Ω are investigated in detail. Finally, it is shown that contractive properties of $${\rho _{{N_T}\left( w \right)}}$$ are equivalent to an inequality for the curvature of the Cowen-Douglas bundle ET. However, we construct examples to show that the contractivity of the homomorphism ρT does not follow, even if $${\rho _{{N_T}\left( w \right)}}$$ is contractive for all w in Ω.
Databáze: OpenAIRE
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