Maximal regularity of discrete and continuous time evolution equations (Q2717713)

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 \(A := T-I\).