Rigidity of contractions on Hilbert spaces

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.

17 pages. The results of this paper are published in Section IV.3 in the author's monograph "Stability of Operators and Operator Semigroups", Springer Verlag

References in corpus (1)

• Category theorems for stable operators on Hilbert spaces

Cited by in corpus (3)

• Recurrent Linear Operators
• On typical properties of Hilbert space operators
• Recurrent strongly continuous operator families on Banach space