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Boundary relations, unitary colligations, and functional models. (English) Zbl 1186.47007
Recently, V. Derkach, S. Hassi, M. Malamud and H. De Snoo [Trans. Am. Math. Soc. 358, No. 12, 5351–5400 (2006; Zbl 1123.47004), Russ. J. Math. Phys. 16, No. 1, 17–60 (2009; Zbl 1182.47026)] have introduced new notions of boundary relations and the corresponding Weyl families. Let \(S\) be a closed symmetric relation in a Hilbert space \(\mathfrak{H}\) and let \(\mathcal{H}\) be an auxiliary Hilbert space. A linear relation \(\Gamma\) from \(\mathfrak{H}^2=\mathfrak{H}\times\mathfrak{H}\) to \(\mathcal{H}^2=\mathcal{H}\times\mathcal{H}\) is said to be a boundary relation for \(S^*\) if (i) \(\mathcal{T}=\text{dom}\,\Gamma\) is dense in \(S^*\) and the identity \((f',g)_{\mathfrak{H}}-(f,g')_{\mathfrak{H}}=(h',k)_{\mathcal{H}}-(h,k')_{\mathcal{H}}\) holds for every \(\{\hat{f},\hat{h}\}\), \(\{\hat{g},\hat{k}\}\in\Gamma\), where \(\hat{f}=\{f,f'\}\), \(\hat{g}=\{g,g'\}\in\mathfrak{H}^2\) and \(\hat{h}=\{h,h'\}\), \(\hat{k}=\{k,k'\}\in\mathcal{H}^2\); (ii) if \(\{\hat{g},\hat{k}\}\in\mathfrak{H}^2\times\mathcal{H}^2\) satisfies the identity in part (i) for every \(\{\hat{f},\hat{h}\}\in\Gamma\), then \(\{\hat{g},\hat{k}\}\in\Gamma\). The Weyl family associated to a boundary relation \(\Gamma\) is the abstract analogue of the classical Titchmarsh-Weyl coefficient or \(m\)-function in Sturm-Liouville theory, and is defined as follows. The Weyl family \(M(\lambda)\), \(\lambda\in\mathbb{C}\setminus\mathbb{R}\), of \(S\) corresponding to the boundary relation \(\Gamma:\mathfrak{H}^2\to\mathcal{H}^2\) is defined by \(M(\lambda)=\{\hat{h}\in\mathcal{H}^2:\{\hat{f}_\lambda,\hat{h}\}\in\Gamma\text{ for\;some } \hat{f}_\lambda=\{f_\lambda,\lambda f_\lambda\}\in\mathcal{H}^2\}\). If the values of \(M(\lambda)\), \(\lambda\in\mathbb{C}\setminus\mathbb{R}\), are operators in \(\mathcal{H}\), then one speaks of a Weyl function instead of a Weyl family. It follows directly from the definition that a Weyl family is a Nevanlinna family, that is, a holomorphic relation-valued function symmetric about the real line and whose values on \(\mathbb{C}_+\) (\(\mathbb{C}_-\)) are maximal dissipative (maximal accumulative, respectively) linear relations. Conversely, it was shown in Derkach et al. [loc. cit.] with the use of the Naimark dilation theorem that each Nevanlinna family can be realized as a Weyl family of some boundary relation in an abstract model space.
In the paper under review, a new approach to boundary relations and their Weyl families is presented. The main idea is that the Weyl family \(M(\lambda)\) and a certain selfadjoint relation \(\tilde{A}\) in \(\mathfrak{H}\times\mathcal{H}\) induced by the boundary relation \(\Gamma:\mathfrak{H}^2\to\mathcal{H}^2\) are connected via the Cayley transform with an operator-valued Schur-class function \(\Theta\) in \(\mathcal{H}\) and a unitary colligation \(U\) in \(\mathfrak{H}\times\mathcal{H}\) with \(\Theta\) as the corresponding transfer function. As a consequence, various subclasses of Nevanlinna families are described and the functional model of Nevanlinna families is constructed from the de Branges-Rovnyak functional model of Schur-class functions.

MSC:
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47A06 Linear relations (multivalued linear operators)
47B25 Linear symmetric and selfadjoint operators (unbounded)
47B50 Linear operators on spaces with an indefinite metric
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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