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The Krein formula for generalized resolvents in degenerated inner product spaces. (English) Zbl 0941.47032

Let \({\mathcal H}\) be a, possibly degenerated, inner product space such that \({\mathcal H}/({\mathcal H}\cap{\mathcal H}^{\perp})\) is a Pontryagin space, and let \(S\) be a symmetric operator in \({\mathcal H}\) with defect index \((1,1)\). If \({\mathcal P}\) is a Pontryagin space containing \({\mathcal H}\) as a singular subspace, and \(A\) is a selfadjoint relation in \({\mathcal P}\) extending \(S\), we call the set of analytic functions \[ [(A- z)^{-1}x,y],\quad x,y\in{\mathcal H}\tag{1} \] a generalized resolvent of \(S\).
In the following, denote by \(\langle\cdot\rangle\) the linear span of the elements between the brackets. If \(A\) is \({\mathcal H}\)-minimal in the sense that \[ \overline{\langle{\mathcal H},(A- z)^{-1}{\mathcal H}|z\in\varrho(A)\rangle}={\mathcal P}, \] the corresponding generalized resolvent is called minimal. In this case, if \({\mathcal P}\) has index \(\kappa\) of negativity \((\kappa= \text{ind}_-{\mathcal P})\), the generalized resolvent is said to have index \(\kappa\).
If \(\tau\) is a complex valued meromorphic function, denote by \(\varrho(\tau)\) its domain of holomorphy. For \(\kappa\in \mathbb{N}_0\) let the set \({\mathcal N}_{\kappa}\) of generalized Nevanlinna functions \(\tau\) be defined as follows: \(\tau\) is meromorphic in \(\mathbb{C}\setminus\mathbb{R}\), \(\tau(\overline z)= \overline{\tau(z)}\) for \(z\in\varrho(\tau)\), and the Nevanlinna kernel \((z,w\in\varrho(\tau))\), \[ N_\tau(z, w)= \begin{cases} {\tau(z)- \tau(\overline w)\over z-\overline w},\quad & z\neq\overline w\\ \tau'(z),\quad & z=\overline w\end{cases} \] has \(\kappa\) negative squares.
If \({\mathcal H}\) is nondegenerated, i.e., a Pontryagin space, Kreĭn’s formula \[ [(A- z)^{-1}x,y]= [(\mathring A- z)^{-1}x,y]- [x,\chi(\overline z)]{1\over \tau(z)+ q(z)} [\chi(z), y]\tag{2} \] establishes a bijective correspondence between the minimal generalized resolvents of index \(\kappa\) of \(S\) and the set \({\mathcal N}_{\kappa- \kappa_0}(\kappa_0= \text{ind}_-{\mathcal H})\) of parameters \(\tau\). Here \(\mathring A\subseteq{\mathcal H}^2\) is a fixed selfadjoint extension of \(S\), \(\chi(z)\) are certain defect elements of \(S\) and \(q(z)\) is a corresponding \(Q\)-function, which means that the relation \(N_q(z,w)= [\chi(z),\chi(w)]\) holds.
In this paper it is considered that \({\mathcal H}\) is actually degenerated, \(\dim{\mathcal H}^0= \Delta>0\). It turns out that the relation (2) still holds, where \(\mathring A\), \(\chi(z)\) and \(q(z)\) have a similar meaning. However, the parameter \(\tau\) does not run through the Nevanlinna class \({\mathcal N}_{\kappa- \kappa_0}\), but through a different set of functions \({\mathcal K}^\Delta_{\kappa- \kappa_0}\) which is defined as follows:
Definition. For \(\nu, \Delta\in \mathbb{N}_0\), denote by \({\mathcal K}^\Delta_\nu\) the set of all complex valued functions \(\tau(z)\), meromorphic in \(\mathbb{C}\setminus\mathbb{R}\), which satisfy \(\tau(\overline z)= \overline{\tau(z)}\) for \(z\in\varrho(\tau)\), and are such that the maximal number of the negative squares of the quadratic forms \((m\in \mathbb{N}_0,z_1,\dots, z_m\in \varrho(\tau))\) \[ Q(\xi_1,\dots, \xi_m; \eta_0,\dots, \eta_{\Delta- 1})= \sum^n_{i,j= 1} N_\tau(z_i, z_j)\xi_i\overline{\xi_j}+ \sum^{\Delta- 1}_{k= 0} \sum^m_{i= 1} \text{Re}(z^k_i\xi_i \overline{\eta_k})\tag{3} \] is \(\nu\).

MSC:

47B50 Linear operators on spaces with an indefinite metric
47B25 Linear symmetric and selfadjoint operators (unbounded)
42A82 Positive definite functions in one variable harmonic analysis
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
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