Hi!

Let $a(x)\geq0$, $x\in R^d$ and $\int_{R^d} a(x) dx=1$. Then the operator $Af = a*f -f$ is bounded on the space of continuous functions on $R^d$ vanishing at infinity and it is a generator of a Markov semigroup, therefore, its spectrum will be only on the left half-plane (and bounded, of course).

If we consider now the Banach space of all bounded continuous functions on $R^d$, then $A$ will be bounded as well, however, what can we say about left half plane. Could it be shown that this operator is dissipative?