Rational functions as new variables

Diana Andrei, Olavi Nevanlinna, Tiina Vesanen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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In multicentric calculus, one takes a polynomial p with distinct roots as a new variable and represents complex valued functions by Cd-valued functions, where d is the degree of p. An application is e.g. the possibility to represent a piecewise constant holomorphic function as a convergent power series, simultaneously in all components of | p(z) | ≤ ρ. In this paper, we study the necessary modifications needed, if we take a rational function r= p/ q as the new variable instead. This allows to consider functions defined in neighborhoods of any compact set as opposed to the polynomial case where the domains | p(z) | ≤ ρ are always polynomially convex. Two applications are formulated. One giving a convergent power series expression for Sylvester equations AX- XB= C in the general case of A, B being bounded operators in Banach spaces with distinct spectra. The other application formulates a K-spectral result for bounded operators in Hilbert spaces.

Original languageEnglish
Article number37
Pages (from-to)1-22
Number of pages22
JournalBanach Journal of Mathematical Analysis
Issue number3
Publication statusPublished - Jul 2022
MoE publication typeA1 Journal article-refereed


  • Functional calculus
  • Rational functions
  • Series expansions


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