Some extensions of Loewner's theory of monotone operator functions.
There are many results devoted to the class of real-valued differentiable functions \(f\) on \((a;b)\subset \mathbb R^1\) such that \(f(A)\leq f(B)\) whenever \(A\) and \(B\) (\(A\leq B\)) are selfadjoint matrices with eigenvalues lying in \((a;b)\). This class coincides with the set of all real-valued differentiable functions \(f\) on \((a;b)\) for which the Hermitian kernel \(K_f(x;y)\geq 0\) \(\forall x,y \in (a;b)\). Moreover, the Hermitian kernel \(K_f\) is nonnegative iff a function \(f\) is the restriction of a Nevanlinna function which has an analytic continuation across \((a;b)\). The paper under review considers a matrix-valued function \(F\) instead of the scalar-valued function \(f\). Several extensions of Loewner's theory of monotone operator functions are obtained. Some theorems on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class are proved. Further generalizations are connected with the monotone functions of an operator argument. The authors introduce and study, by analogy with classical Nevanlinna theory, a notion of \(k\)-monotonicity for such operator functions. It should be noted that there are corresponding results for Stieltjes functions.
scientific article; zbMATH DE number 1747014