Curvature inequalities and extremal operators. (English) Zbl 1429.30017

For a bounded domain \(\Omega\subset\mathbb{C}^m\), let \(\Omega^*=\{\overline{z}:z\in\Omega\}\). An \(m\)-tuple \(\mathbf{T}=(T_1,T_2,\ldots,T_m)\) of commuting bounded operators on a complex separable Hilbert space \(\mathcal{H}\) is said to be in \(B_n(\Omega^*)\) if:
\(\dim\big(\bigcap_{k=1}^m\ker(T_k-w_kI)\big)=n\) for each \(w=(w_1,w_2,\ldots,w_m)\in\Omega^*\);
the operator \(\mathcal{D}_{\mathbf{T}-w\mathbf{I}} =\bigoplus_{k=1}^m(T_k-w_kI)\), \(w\in\Omega^*\), has closed range in the Hilbert space \(\bigoplus_{k=1}^m\mathcal{H}\);

For any commuting tuple of operators \(\mathbf{T}\) in \(B_n(\Omega^*)\), there exists a rank-\(n\) holomorphic Hermitian vector bundle \(E_T\) over \(\Omega^*\), \[ E_T:=\Big\{(w,v)\in\Omega^*\times\mathcal{H}:\ v\in\bigcap_{k=1}^m\ker(T_k-w_kI)\Big\}. \] For \(w\in\Omega\), properties of the curvature \(\mathcal{K}(w)\) of the vector bundle \(E_T\) are investigated. For contractive commuting tuples of operators \(\mathbf{T}\in B_n(\Omega^*)\), a curvature inequality is established. The properties of the extremal operators transforming the curvature inequality into the equality are studied. For the open unit disc \(\mathbb{D}\), the unitary equivalence of a contraction \(T\in B_1(\mathbb{D})\) and the backward shift operator \(U_+^*\) is proved under two additional conditions.


30C40 Kernel functions in one complex variable and applications
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A25 Spectral sets of linear operators
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