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Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations. (English) Zbl 1223.47012

The authors study Schur-class multipliers on the Arveson space \(\mathcal{H}(K_S)\). The latter is the reproducing kernel Hilbert space associated with the reproducing kernel \(k_d(\lambda,\zeta)=1/(1-\langle \lambda,\zeta\rangle)\) on the unit ball \(\mathbb{B}^d\subset\mathbb{C}^d\). If \(\mathcal{U}\) and \(\mathcal{Y}\) are Hilbert spaces, then \(\mathcal{H}_\mathcal{U}(k_d)=\mathcal{H}(k_d)\otimes\mathcal{U}\) and \(\mathcal{H}_\mathcal{Y}(k_d)=\mathcal{H}(k_d)\otimes\mathcal{Y}\) are vector-valued Arveson spaces, and the multivariable Schur class \(\mathcal{S}_d(\mathcal{U,Y})\) consists of \(\mathcal{L(U,Y)}\)-valued functions \(S\) on \(\mathbb{B}^d\) which are analytic and such that the multiplication operator \(M_S:\mathcal{H}_\mathcal{U}(k_d)\to\mathcal{H}_\mathcal{Y}(k_d)\) is contractive. A multivariable analogue of the classical de Branges–Rovnyak space for the unit ball case is the reproducing kernel Hilbert space \(\mathcal{H}(K_S)\) associated with the reproducing kernel \(K_S(\lambda,\zeta)=(I_\mathcal{Y}-S(\lambda)S(\zeta)^*)k_d(\lambda,\zeta)\) determined by \(S\in\mathcal{S}_d(\mathcal{U,Y})\).
In general, the space \(\mathcal{H}(K_S)\) is not invariant under the operators \(M_{\lambda_j}^*\), \(j=1,\dots,d\), where \(M_{\lambda_j}:f(\lambda)\mapsto \lambda_jf(\lambda)\) are multipliers on \(\mathcal{H}_\mathcal{Y}(k_d)\). The authors show that this invariance is equivalent to the existence of a transfer-function realization \(S(\lambda)=D+C(I_{\mathcal X}-\lambda_1A_1-\cdots -\lambda_dA_d)^{-1}(\lambda_1B_1+\cdots +\lambda_dB_d)\) such that the connecting operator (colligation) \(U\) has adjoint \(U^*\) which is isometric on a certain natural subspace (\(U\) is said to be weekly coisometric in this case) and, in addition, the operators \(A_1,\dots,A_d\) pairwise commute. Moreover, the model state space \(\mathcal{X}\) for such a realization is \(\mathcal{H}(K_S)\) and \(A_j=M_{\lambda_j}^*| \mathcal{H}(K_S)\), \(j=1,\dots,d\). It is also shown in the paper that inner multipliers, i.e., partial isometries \(M_S:\mathcal{H}_\mathcal{U}(k_d)\to\mathcal{H}_\mathcal{Y}(k_d)\), can be characterized as those for which there is a weekly coisometric realization of \(S\) where \(A_1,\dots, A_d\) commute and have an additional stability property. In addition, the paper provides a realization-theoretic proof of the Beurling–Lax theorem for \(\mathcal{H}_\mathcal{Y}(k_d)\) which characterizes \(M_{\lambda_j}\)-invariant subspaces.

MSC:

47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A15 Invariant subspaces of linear operators
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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