Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2016; 53(4): 895-908

Printed July 1, 2016

https://doi.org/10.4134/JKMS.j150334

Copyright © The Korean Mathematical Society.

Normal eigenvalues in evolutionary process

Dohan Kim, Rinko Miyazaki, Toshiki Naito, and Jong Son Shin

Seoul National University, Shizuoka University, The University of Electro-Communications, Hosei University

Abstract

Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $C_0$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $P_{\tau}(X)$ associated with the $\tau$-periodic evolutionary process on a Banach space $X$, where $P_{\tau}(X)$ stands for the space of all $\tau$-periodic continuous functions mapping ${\mathbb R}$ to $X$.

Keywords: $C_0$-semigroup, evolution semigroup, monodromy operator, normal eigenvalue, order of pole, ascent

MSC numbers: Primary 47A10, 47D06; Secondary 35K05