I was wondering if there were any easy results regarding the spectral radius of a Hilbert-Schmidt operator. The Hilbert-Schmidt integral operator is defined by $$ Tf(x) = \int k(x,y) f(y) \: dy . $$ I'm guessing there is not an easy general form for the spectral radius of this operator, since in the case I'm dealing with, Gelfand's formula yields something of the form $$\lim_{n \to \infty} \Big|\Big| \int k^n(x,y) g(y) \: dy \Big|\Big|^{1/n}$$ where $k^n(x,y)$ can be a rather complicated and varied expression. Nevertheless, a simple result that holds in general would be very useful.
Thanks for any help!