Let T be an compact, positiv, linear operator on any hilbert space. I know from the SVD that there exists some eigenvalues $\lambda_i$ and eigenfunctions $\phi_i$ such that
$$ Tf=\sum_{i=0}^\infty\lambda_i\langle f, \phi_i \rangle \phi_i. $$
So i can use spectral properties as $||T^r||\leq \max_i{\lambda_i^r} $ and moreover since T is positiv $||(I-T)^r||\leq (1 - \max_i{\lambda_i})^r $ for any integer $r > 0$ if $\max_i{\lambda_i^r}<1$.
Now I need to do the same for an operator $T:C\rightarrow C$, where C denotes the space of continuous functions on $[0,1]$.
More precisely let K be a bounded continous positiv kernel $|K(x,y)|< 1$ and consider the integral operator $$T:C\rightarrow C; Tf=\int_{\mathbb{[0,1]}} K(.,y)f(y)dy.$$
Is there any way to bound $$\sup_{||f||_{\infty}<1}||(I-T)^r f||_\infty<1$$ or maybe $$ \sup_{||f||_{\infty}<1}||(I-T)^r f||_{L^2[0,1]}<1 \,\,\,??? $$