Rigidity of contractions on Hilbert spaces

TL;DR

This paper investigates the long-term behavior of contractive operators and semigroups on Hilbert spaces, revealing typical rigidity properties and convergence patterns of their powers.

Contribution

It introduces the notion of rigidity to analyze asymptotic behavior, showing that typical contractions exhibit specific strong and weak limit properties.

Findings

Typical contractions have the unit circle times the identity in their strong limit set.

Powers of typical contractions converge weakly to zero along sequences with density one.

The results extend to isometric and unitary $C_0$-(semi)groups.

Abstract

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times the identity operator in the strong limit set of its powers, while $T^{n_j}$ converges weakly to zero along a sequence $\{n_j\}$ with density one. The continuous analogue is presented for isometric ang unitary $C_0$-(semi)groups.