Operator-Valued Poisson Kernels and Standard Models in Several Variables

Let H be a complex Hilbert space, and let L(H) be the algebra of all bounded linear oeprators acting on H. For each T ∈ L(H) we denote by σ(T the spectrum of T. Let D ⊂ ℌ be the open unit disc. If T ∈ L(H) is such that σ(T) ⊂ D, then for each analytic polynomial f (more generally, for each f holomorphic in D and continuous on $$\bar{\mathbb{D}}$$ we have (see [Vas]) (1.1) $$f\left( T \right) = \int_{{\partial \mathbb{D}}} {f\left( w \right){{{\left( {I - w{{T}^{*}}} \right)}}^{{ - 1}}}\left( {I - {{T}^{*}}T} \right){{{\left( {I - \bar{w}T} \right)}}^{{ - 1}}}d\sigma \left( w \right)}$$ where σ is the normalized Lebesgue measure on ∂D. If, moreover,∥T∥ ≤ 1, then (1.2) $$P\left( {T,w} \right) = {{\left( {I - w{{T}^{*}}} \right)}^{{ - 1}}}\left( {I - {{T}^{*}}T} \right){{\left( {I - wT} \right)}^{{ - 1}}},w \in \mathbb{D}$$ is positive for all w ∈ D