Spectrum of composition operators on \mathcal S(\mathbb R) with polynomial symbols

2018/10/31 by Carmen Fernández, Fernández, Carmen, Antonio Galbis +3
Mathematics · #46E10 (Primary) 47B33 #47A10 #47A35 (Secondary) #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1810.13208

openalex publication_date 2018/10/31 · openalex created_date 2018/11/09 · openalex updated_date 2026/07/28

Abstract

We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

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