Glasnik Matematicki, Vol. 35, No.1 (2000), 75-87.

OPERATOR REPRESENTATIONS OF N_∞⁺-FUNCTIONS IN A MODEL KREIN SPACE L_σ²

Andreas Fleige

Abstract.   We introduce the class N_∞⁺ of all complex functions Q such that Q₊(z) := z · Q(z) is a Nevalinna function. If 0 ∈ D(Q) and lim_{y → ∞} Q(iy) = 0 we prove an integral representation Q(z) = ∫_-∞^∞ 1/(t-z) dσ(t) with a nonmonotonic function σ. If in particular Q₊ is an R₁-function we obtain an operator representation Q(z) = [(A - z) ^-1 F_-, F_-]_σ with a selfadjoint, nonnegative and boundedly invertible multiplication operator A in the model Krein space (L_σ², [.,.]_σ) and an element F_- ∈ L_σ². The nonsingularity of the critical point infinity of A makes this representation unique up to a Krein space isomorphism.

1991 Mathematics Subject Classification.   47B50, 30E20, 46C20, 47B38.