Contractivity, complete contractivity and curvature inequalities

2014/10/28 by Gadadhar Misra, Avijit Pal, Misra, Gadadhar +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1410.7493

The material in this paper is taken from the PhD thesis, arXiv:1410.6394, of the second author

arxiv created 2015/01/17 · arxiv updated 2015/01/20

Abstract

For any bounded domain Ω in \mathbb C^m, let \mathrm B₁(Ω) denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple \boldsymbol T in the Cowen-Douglas class \mathrm B₁(Ω), let N_{\boldsymbol T}(w) denote the restriction of \boldsymbol T to the subspace ∩_i,j=1^mker(T_i-w_iI)(T_j-w_jI). This commuting m-tuple N_{\boldsymbol T}(w) of m+1 dimensional operators induces a homomorphism ρ__ N_{\boldsymbol T}(w) of the polynomial ring P[z₁, ..., z_m], namely, ρ__ N_{\boldsymbol T}(w)(p) = p (N_{\boldsymbol T}(w) ), p∈ P[z₁, ..., z_m]. We study the contractivity and complete contractivity of the homomorphism ρ__ N_{\boldsymbol T}(w). Starting from the homomorphism ρ__ N_{\boldsymbol T}(w), we construct a natural class of homomorphism ρ_ _N^(λ)(w), λ>0, and relate the properties of ρ_ _N^(λ)(w) to that of ρ__ N_{\boldsymbol T}(w). 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 ρ__ N_{\boldsymbol T}(w) is equivalent to an inequality for the curvature of the Cowen-Douglas bundle E_{\boldsymbol T}.

Citations

• Maximal regularity for non-autonomous equations with measurable\n dependence on time

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