An Example in the Theory of Bisectorial Operators

An unbounded operator is said to be bisectorial if its spectrum is contained in two sectors lying, respectively, in the left and right half-planes and the resolvent decreases at infinity as $1/\lambda$. It is known that, for any bounded function $f$, the equation $u'-Au=f$ with bisectorial operator $A$ has a unique bounded solution $u$, which is the convolution of $f$ with the Green function. An example of a bisectorial operator generating a Green function unbounded at zero is given.