Advanced Search

Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and $\left|\right|T^{n+1}-Tⁿ\left|\right|\asymp \left(n\ge 1\right)$. This answers Zemánek’s question on the time regularity property.

In this note we give a negative answer to Zem�nek’s question (1994) of whether it always holds that a Cesàro bounded operator $T$ on a Hilbert space with a single spectrum satisfies ${lim}_{n\to \infty }\parallel T^{n+1}-T^n\parallel =0.$

We prove that some regularity conditions on unbounded representations of topological abelian semigroups with countable spectral conditions induce a certain stability result extending the well-known Arendt-Batty-Lyubich-Vũ theorem.