Lifting of operators and prescribed numbers of negative squares
The authors extend a problem of dilation theory from the case of definite inner product spaces to that of Pontryagin spaces. Using the Hilbert spaces \({\mathcal K}_ 1\), \({\mathcal K}_ 2\), \({\mathcal K}'_ 1\) and \({\mathcal K}'_ 2\), and the corresponding symmetries \(J_ 1\), \(J_ 2\), \(J'_ 1\) and \(J'_ 2\), they consider \(\tilde {\mathcal K}_ i={\mathcal K}_ i\oplus {\mathcal K}'_ i\) and \(\tilde J_ i=J_ i\oplus J'_ i\), \(i=1,2\). If \(T\in {\mathcal L}({\mathcal K}_ 1,{\mathcal K}_ 2)\) is such that the negative signature of \(J_ 1-T^*J_ 2T\) is a given cardinal k, and \(\bar k\) is another given cardinal, the problem (denoted (*)) is to give a description of all operators \(\tilde T\in {\mathcal L}(\tilde {\mathcal K}_ 1,\tilde {\mathcal K}_ 2)\) such that \(P_{{\mathcal K}_ 2}^{\tilde {\mathcal K}_ 2} \tilde T | {\mathcal K}_ 1=T\) and \(k^-(\tilde J_ 1-\tilde T^*\tilde J_ 2\tilde T)=\bar k\). The answer is mainly contained in the following theorem: (i) If (*) has solutions, then \[ \bar k\geq \max \{k-\bar k(\tilde {\mathcal K}_ 2)+k^-({\mathcal K}_ 2),\quad k^-(\tilde {\mathcal K}_ 1)-k^- (\tilde {\mathcal K}_ 2)\}. \] (ii) The formula \[ T=\left( \begin{matrix} T\\ \Gamma_ 2D_ T\end{matrix} \begin{matrix} D_{T^*}\Gamma_ 1\\ -\Gamma_ 2L^*_ TJ_{T^*}\quad \Gamma_ 1+D_{\Gamma^*_ 2}\Gamma D_{\Gamma_ 1}\end{matrix} \right) \] establishes a one-to-one correspondence between the set of all solutions of (*) with \(\bar k=k- k^-(\tilde {\mathcal K}_ 2)+k^-({\mathcal K}_ 2)\geq k^-(\tilde {\mathcal K}_ 1)-k^-(\tilde {\mathcal K}_ 2)\) and the triplets \(\{\Gamma_ 1,\Gamma_ 2,\Gamma \}\), where \(\Gamma_ 1\in {\mathcal L}({\mathcal K}'_ 1,{\mathcal D}_{T^*})\) and \(\Gamma_ 2\in {\mathcal L}({\mathcal D}_ T,{\mathcal K}'_ 2)\), \(\Gamma_ 1\) and \(\Gamma^{\#}_ 1\) being J-contractions, while \(\Gamma\in {\mathcal L}({\mathcal D}_{\Gamma_ 1},{\mathcal D}_{\Gamma^*_ 2})\) is a Hilbert space contraction.
scientific article; zbMATH DE number 4028222