Translation semigroups and their linearizations on spaces of integrable functions

Annette Grabosch
1989 Transactions of the American Mathematical Society  
Of concern is the unbounded operator A/ = f' with nonlinear domain D(A(f)} which is considered on the Banach space E of Bochner integrable functions on an interval with values in a Banach space F. Under the assumption that cJ> is a Lipschitz continuous operator from E to F, it is shown that A generates a strongly continuous translation semigroup (T(t))t>o. For linear operators cJ> several properties such as essential-compactness, positivity, and irreducibility of the semigroup (T(t))t>o
more » ... g on the operator cJ> are studied. It is shown that if F is a Banach lattice with order continuous norm, then (TI1 (t))t~O is the modulus semigroup of (T(t))t>o. Finally spectral properties of A are studied and the spectral bound s(A is linear and to a local stability result in the case where cJ> is Frechet differentiable.
doi:10.1090/s0002-9947-1989-0974781-2 fatcat:j5aldaczhjdhnhkebokugiw4vq