On the structure of semigroups on \(L_{p}\) with a bounded \(H^{\infty}\)-calculus
The author considers bounded analytic semigroups on a reflexive \(L_p\)-space and shows the following converse of Weis' result, generalizing Le Merdy's theorem from the Hilbert space case to an \(L_p\)-setting: for a semigroup \((T (t ))_{t \geq 0}\) on \(L_p\) \(( p \in (1, \infty))\) whose negative generator has a bounded \(H^\infty (\Sigma_{\theta})\)-calculus for some \(\theta < {\pi}/{2}\) there exists a bounded analytic semigroup \(( R( z ))\) on some bigger \(L_p\)-space, which is positive and contractive on the real line, \(( R( z ))\)-invariant closed subspaces \(N \subset M\) and an isomorphism \(S\) from \(L_p\) onto \(M / N\) such that \(T (t )= S^{-1}\hat R(t )S\) for all \(t \geq 0\), where \((\hat R(t ))_{t \geq 0}\) is the induced semigroup on \(M / N\).NEWLINENEWLINEAn analogous result is also obtained for semigroups on subspace quotients of general unconditional martingale difference (UMD)-Banach lattices.NEWLINENEWLINESeveral useful remarks illustrating the problem on general (UMD)-Banach spaces and lattices are included. Moreover, some open problems associated with the subject are proposed.
scientific article; zbMATH DE number 6355604
scientific article