Decomposing and twisting bisectorial operators

Wolfgang Arendt¹ ; Alessandro Zamboni²

¹ Institute of Applied Analysis University of Ulm Helmholtzstr. 18 89081 Ulm, Germany
² Dipartimento di Matematica Università degli Studi di Parma via G. P. Usberti 53//A 43100 Parma, Italy

Studia Mathematica, Tome 197 (2010) no. 3, pp. 205-227

Résumé

Bisectorial operators play an important role since exactly these operators lead to a well-posed equation $u'(t)=Au(t)$ on the entire line. The simplest example of a bisectorial operator $A$ is obtained by taking the direct sum of an invertible generator of a bounded holomorphic semigroup and the negative of such an operator. Our main result shows that each bisectorial operator $A$ is of this form, if we allow a more general notion of direct sum defined by an unbounded closed projection. As a consequence we can express the solution of the evolution equation on the line by an integral operator involving two semigroups associated with $A$.

Détail

DOI : 10.4064/sm197-3-1

Keywords: bisectorial operators play important role since exactly these operators lead well posed equation entire line simplest example bisectorial operator obtained taking direct sum invertible generator bounded holomorphic semigroup negative operator main result shows each bisectorial operator form allow general notion direct sum defined unbounded closed projection consequence express solution evolution equation line integral operator involving semigroups associated