On Matrix-Valued Stieltjes Functions with an Emphasis on Particular Subclasses

TL;DR

This paper explores special classes of matrix-valued functions that are holomorphic outside a real interval, generalizing known classes and relating them to truncated matricial Stieltjes problems, with focus on integral representations and Moore-Penrose inverses.

Contribution

It introduces and characterizes new subclasses of matrix-valued Stieltjes functions, extending existing theories and analyzing their integral representations and Moore-Penrose inverses.

Findings

Characterizations via integral representations are provided.

Relations to truncated matricial Stieltjes problems are established.

Analysis of Moore-Penrose inverse properties is included.

Abstract

The paper deals with particular classes of $q\times q$ matrix-valued functions which are holomorphic in $\mathbb{C}\setminus[\alpha,+\infty)$, where $\alpha$ is an arbitrary real number. These classes are generalizations of classes of holomorphic complex-valued functions studied by Kats and Krein [17] and by Krein and Nudelman [19]. The functions are closely related to truncated matricial Stieltjes problems on the interval $[\alpha,+\infty)$. Characterizations of these classes via integral representations are presented. Particular emphasis is placed on the discussion of the Moore-Penrose inverse of these matrix-valued functions.