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On the Kreĭn-Langer integral representation of generalized Nevanlinna functions. (English) Zbl 1059.47014

The paper is devoted to an analogue of the Kreǐn-Langer integral representation for matrix-valued generalized Nevanlinna functions \(v\in N_\kappa\) \((\kappa\geq 0)\). The class \(N_\kappa\) consists of the meromorphic \(m\times m\) matrix functions \(v(z)\) on the union \(\mathbb{C}_+\cup\mathbb{C}_-\) of the upper and lower half-planes such that \(v(\bar{z})^*=v(z)\) and the kernel \[ K(z,\zeta)=\frac{v(z)-v(\zeta)^*}{z-\bar{\zeta}} \] has \(\kappa\) negative squares. The latter condition means that for any finite set of points \(z_1,\ldots,z_n\) in the domain of analyticity of \(v(z)\) and vectors \(c_1,\ldots,c_n\) in \(\mathbb{C}^m\), the matrix \([c_kK(z_j,z_k)c_j]_{j,k=1}^n\) has at most \(\kappa\) negative eigenvalues, and at least one such matrix has exactly \(\kappa\) negative eigenvalues (counting multiplicity).
The Nevanlinna integral representation for \(v\in N_\kappa\) was established in the scalar case by M. G. Kreǐn and H. Langer [ibid. 77, 187–236 (1977; Zbl 0412.30020)] and for the matrix-valued functions by K. Daho and H. Langer [Math. Nachr. 120, 275-294 (1977; Zbl 0567.47030)].
The authors present a new version of the Kreǐn-Langer integral representation for matrix-valued generalized Nevanlinna functions \(v\in N_\kappa\), which is closer to the scalar case. They also obtain a Stieltjes inversion formula for matrix functions \(v\in N_\kappa\) and characterize two subclasses of \(N_\kappa\) in terms of their Kreǐn-Langer integral representations.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
30D99 Entire and meromorphic functions of one complex variable, and related topics
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
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