Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set
Let \(\Omega\) be a bounded domain in \(\mathbb{C}^m\). The Cowen-Douglas class consists of \(m\)-tuples of commuting bounded operators \(\mathbf{T}=(T_1,\dots,T_m)\) on a Hilbert space \(\mathcal{H}\) such that for \(w=(w_1,\dots,w_m)\in\Omega\), the dimension of the joint kernel \(\cap_{k=1}^m\ker(T_k-w_kI)\) is equal to \(1\); for \(w\in\Omega\) and \(h\in\mathcal{H}\), the operator \(D_{\mathbf{T}-wI}:\mathcal{H}\to\mathcal{H}\oplus\dots\oplus\mathcal{H}\) defined by \(D_{\mathbf{T}-wI}=((T_1-w_1I)h,\dots,(T_m-w_mI)h))\) has closed range; the closed linear span of \(\left\{\cap_{k=1}^m\ker (T_k-w_kI):w\in\Omega\right\}\) coincides with \(\mathcal{H}\). For an \(m\)-tuple \(\mathbf{T}\in B_1(\Omega)\), let \(N_{\mathbf{T}}(w)\) denote the restriction of \(\mathbf{T}\) to the subspace \(\mathcal{N}(w):=\cap_{i,j=1}^m\ker (T_i-w_iI)(T_j-w_jI)\). The authors study the contractivity and complete contractivity of homomorphisms of the polynomial ring \(P[z_1,\dots,z_m]\) defined by \(\rho_{N_{\mathbf{T}}(w)}(p)=p(N_{\mathbf{T}}(w))\) for \(p\in P[z_1,\dots,z_m]\). Some explicit examples arising from multiplication operators on the Bergman space of \(\Omega\) are considered in detail. For operators in \(B_1(\Omega)\), there exists a holomorphic choice \(\gamma(w)\) of eigenvalues at \(w\). The map \(\gamma\) defines a holomorphic Hermitian vector bundle \(E_{\mathbf{T}}\) on \(\Omega\). One can define the curvature \(\mathcal{K}_{\mathbf{T}}(w):=-\left(\left( \frac{\partial^2}{\partial w_i\partial\overline{w_j}}\log\|\gamma(w)\|^2 \right)\right)_{i,j=1}^m\). It is shown that contractive properties of \(\rho_{N_{\mathbf{T}}(w)}\) are equivalent to an inequality for the curvature of the Cowen-Dougles bundle \(E_{\mathbf{T}}\).
scientific article; zbMATH DE number 7008548
scientific article