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Maximal regularity of discrete and continuous time evolution equations. (English) Zbl 0981.39009
The author considers the maximal regularity problem for the discrete time evolution equation $$u_{n+1}- T u_n=f_n$$ for all $$n \in N_0$$, $$u_0=0$$, where $$T$$ is a bounded operator on a UMD space $$X$$. The maximal regularity of $$T$$ is characterized by two types of conditions: the first one by $$R$$-boundedness properties of the discrete time semigroup $$\{T^n\}_{n \in N_0}$$ and for the resolvent $$R(\lambda, T)$$; the second one by the maximal regularity of the continuous time evolution equation $$u'(t)- A u(t)=f(t)$$ for all $$t >0$$, $$u(0)=0$$, where $$A := T-I$$.

##### MSC:
 39A12 Discrete version of topics in analysis 42A45 Multipliers in one variable harmonic analysis 47D06 One-parameter semigroups and linear evolution equations
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