Por defecto: 

Let S(R) denote the Fréchet space consisting of those smooth functions on the real line whose derivatives are rapidly decreasing. We discuss the dynamics and spectra of the operators on S(R) defined after composition with a fixed non constant polynomial P. Such an operator is mean ergodic if and only if the polynomial P lacks fixed point. It turns out that the dynamical properties are related to the spectrum. In fact, the composition operator is mean ergodic if and only if its spectrum reduces to {0}. A complete description of the spectrum of the composition operator is obtained for quadratic polynomials and also for cubic polynomials with positive leading coefficient. To be more precise, for a quadratic polynomial P the spectrum reduces to {0} when P lacks fixed point, it coincides with the closed unit disc when P admits a unique fixed point and it coincides with the complex plane in the case that P has two different fixed points. For a cubic polynomial with positive leading coefficient, the spectrum always contains the complex plane except possibly the origin unless the polynomial admits a fixed point with multiplicity three in which case the spectrum coincides with the closed unit disc with the origin deleted.

 

The results of the talk are contained in the following papers:

 

C. Fernández, A. Galbis, E. Jordá; Dynamics and spectra of composition operators on the Schwartz space, J. Funct. Anal. 274 (2018), 3503-3530.

C. Fernández, A. Galbis, E. Jordá; Spectrum of composition operators on S(R) with polynomial symbols. arXiv:1810.13208v1

 

Keywords: Composition operators; space of rapidly decreasing functions; spectrum; mean ergodic

Composition operators; space of rapidly decreasing functions; spectrum; mean ergodic