Discrete time analytic semigroups and the geometry of Banach spaces (Q1402912)

The present paper deals with the Cauchy problem of the form \[ \begin{cases} u_{n+1} -Tu_ n=f_ n, \\ u_ 0=0, \end{cases} \] where \(\{u_ n\}_{n=0}^{\infty}\) are unknown elements of a Banach space \(X,\) \(T\) is a bounded linear operator on \(X\) and the elements \(\{f_ n\}_{n=0}^{\infty}\) are fixed. If \(Y(X)\) is an \(X\)-valued sequence space, the problem is said to have \(Y\)-discrete maximal regularity if \((u_{n+1}-u_n)_{n\geq 0}\) belongs to \(Y(X)\) whenever \(\{f_n\}_{n\geq 0}\) belongs to \(Y(X).\) The author studies \(\ell^2\)-discrete maximal regularity and relations with other properties of the semigroup \((T^n)_{n\geq 0}.\) Also, it is shown that the existence of a copy of \(c_0\) in \(X\) is related to the existence of particular semigroups with \(\ell^\infty\)-discrete maximal regularity.