Decomposing and twisting bisectorial operators
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 $A$.