Kreiss bounded and uniformly Kreiss bounded operators

0542394 - MÚ 2022 RIV ES eng J - Journal Article
Bonilla, A. - Müller, Vladimír
Kreiss bounded and uniformly Kreiss bounded operators.
Revista Mathématica Complutense. Roč. 34, č. 2 (2021), s. 469-487. ISSN 1139-1138. E-ISSN 1988-2807
R&D Projects: GA ČR(CZ) GX20-31529X
Institutional support: RVO:67985840
Keywords : Cesàro mean * Kreiss boundedness * mean ergodic
OECD category: Pure mathematics
Impact factor: 1.009, year: 2021
Method of publishing: Limited access
https://doi.org/10.1007/s13163-020-00355-x

If T is a Kreiss bounded operator on a Banach space, then ‖ Tn‖ = O(n). Forty years ago Shields conjectured that in Hilbert spaces, ‖Tn‖=O(n). A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss bounded operators. More precisely, for every ε> 0 there exists a uniformly Kreiss bounded operator T on a Hilbert space such that ‖ Tn‖ ∼ (n+ 1) 1 - ε for all n∈ N. On the other hand, any Kreiss bounded operator on Hilbert spaces satisfies ‖Tn‖=O(nlogn). We also prove that the residual spectrum of a Kreiss bounded operator on a reflexive Banach space is contained in the open unit disc, extending known results for power bounded operators. As a consequence we obtain examples of mean ergodic Hilbert space operators which are not Kreiss bounded.
Permanent Link: http://hdl.handle.net/11104/0319811