On a connection between Naĭmark's dilation theorem, spectral representations, and characteristic functions (Q2898364)

Let \(U\) be a unitary operator on a Hilbert space \(\mathfrak H\), with its (ordinary) spectral measure \(E:= E(\Delta)\), that is, \(E(\Delta)\) is an orthogonal projection for any Borel set \(\Delta \subset \mathrm{T}\) and, for any \(h_{1}, h_{2} \in \mathfrak H\), NEWLINE\[NEWLINE \langle (U+ zI) (U-zI)^{-1}h_{1}|h_{2}\rangle = \int_{\mathrm{T}} \frac{\xi+z}{\xi-z} d \langle (E(\xi))h_{1} | h_{2} \rangle .NEWLINE\]NEWLINE The author shows that if \(\mathfrak K\) is a closed subspace of \(\mathfrak H\), \(P_{\mathfrak K} : \mathfrak H \rightarrow \mathfrak K\) is the orthogonal projection onto \(\mathfrak K\), and the generalized spectral measure \(B(\Delta)=P_{\mathfrak K} E(\Delta)\), then, for any \(k_{1}, k_{2} \in \mathfrak K\), NEWLINE\[NEWLINE \langle (\widetilde{U}+ \Theta_{S}(z)) (\widetilde{U}-\Theta_{S}(z)I)^{-1}k_{1}| k_{2}\rangle = \int_{\mathrm{T}} \frac{\xi+z}{\xi-z} d \langle (B(\xi))k_{1} | k_{2} \rangle ,NEWLINE\]NEWLINE where \(\widetilde{U}:U^{*}(\mathfrak K) \rightarrow \mathfrak K\) is the restriction of \(P_{\mathfrak K}\) on \(U^{*}(\mathfrak K)\) and \(\Theta_{S}(z)\) is the characteristic function corresponding to the operator \(S:=(I-P_{\mathfrak K})U\) in the sense of B. Sz.-Nagy and C. Foias. For one-dimensional \(\mathfrak K\), the last representation reduces to a relation used by \textit{D. Clark} in [J. Anal. Math. 25, 169--191 (1972; Zbl 0252.47010)].NEWLINENEWLINEConversely, given a Hilbert space \( \mathfrak K\) with a generalized spectral measure \(B:=B(\Delta)\) (\(\Delta \subset \mathrm{T}\)), the author defines the analytic contracted-valued function \(\Theta (z) = (F(z)-I)(F(z)+I)^{-1}\), where \(\langle F(z)k_{1}| k_{2}\rangle = \int_{\mathrm{T}} \frac{\xi+z}{\xi-z} d \langle (B(\xi))k_{1} | k_{2} \rangle\), and considers a completely non-unitary contraction in a Hilbert space whose characteristic function coincides with \(\Theta (z)\). This leads to a new Hilbert space \(\mathfrak H \supseteq \mathfrak K\), a spectral measure \(E:= E(\Delta)\) (\(\Delta \subset \mathrm{T}\)) with orthogonal projections \( E(\Delta)\) such that \(B(\Delta)= P_{\mathfrak K} E(\Delta)\), that is, the classical Naĭmark's dilation theorem is given.NEWLINENEWLINEThe same approach is used to describe the spectrum of all unitary rank-one perturbations of a given partial isometry.