### A class of generalized Ornstein transformations with the weak mixing property

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Let T be a linear operator on a Banach space X with $sup\u2099\left|\right|T\u207f/{n}^{w}\left|\right|<\infty $ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) ${n}^{-1}{\sum}_{k=0}^{n-1}{T}^{k}$ converges uniformly; (ii) $cl(I-T)X=z\in X:li{m}_{n}{\sum}_{k=1}^{n}{T}^{k}z/kexists$.

Let $(L,|\left|\xb7\right|{|}_{L})$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T=T\left(u\right):u=(u\u2081,...,{u}_{d}),{u}_{i}>0,1\le i\le d$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.

Let L be a Banach lattice of real-valued measurable functions on a σ-finite measure space and T=${T}_{t}$: t < 0 be a strongly continuous semigroup of positive linear operators on the Banach lattice L. Under some suitable norm conditions on L we prove a general differentiation theorem for superadditive processes in L with respect to the semigroup T.

Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.

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 ${lim}_{n\to \infty}\parallel {T}^{n+1}-{T}^{n}\parallel =0.$