Dynamics of unbounded linear operators

TL;DR

This paper extends concepts from bounded linear dynamics to unbounded operators, establishing hypermixing criteria and demonstrating hypermixing behavior in several key unbounded operators across different function spaces.

Contribution

It introduces a hypermixing criterion for unbounded operators and proves hypermixing for specific operators like the derivative, Laplacian, and weighted translations.

Findings

Derivative operator in H^2 is hypermixing.

Laplacian operator in L^2(Ω) is hypermixing.

Weighted translations in L_p(0,∞) and C_0[0,∞) are hypermixing.

Abstract

We apply the well-known and also the newly introduced notions from bounded linear dynamics to unbounded linear operators. We present a hypermixing criterion similar to that given for bounded linear operators and then we show that the derivative operator in $H^2$, the Laplacian operator in $L^2(\Omega)$, and all unbounded weighted translations in $L_p(0,\infty)$ and $C_0[0,\infty)$ are hypermixing.