## Abstract

In multicentric calculus, one takes a polynomial p with distinct roots as a new variable and represents complex valued functions by C^{d}-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 language | English |
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Article number | 37 |

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Banach Journal of Mathematical Analysis |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2022 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Functional calculus
- Rational functions
- Series expansions