Spectrum of convolution operator on bounded continuous functions (on R^d)

Question

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?

Answer 1

If it's dissipative on $C_0({\bf R}^d)$ then it should be dissipative on $C_b({\bf R}^d)$ too.

First check this in the case that $a$ has bounded support. The point is that if $a$ is supported on a ball of radius $r$ about the origin then $Af(x)$ only depends on values of $f(y)$ for $|x - y| < r$. So if $f \in C_b({\bf R}^d)$ and $x \in {\bf R}^d$ then for $R \gg r$ the function $$g(y) = \cases{\big(1 - \frac{|x - y|}{R}\big)f(y)&if $|x - y| < R$\cr0&if $|x - y| \geq R$\cr}$$ will satisfy $Ag(y) \approx (1 - \frac{|x-y|}{R})Af(y)$ for $|x - y| < R$. Thus $(\lambda I - A)g(y) \approx (1 - \frac{|x-y|}{R})(\lambda I - A)f(y)$ and the dissipation inequality for $g$ will imply approximately the same inequality for $f$ at $x$.

For general $a$, approximate by $a$ with bounded support. This part is easy.