Does anybody know a reference for the following theorem?

Let $G \subset \mathbb{R}^m$ be open and of finite measure and $T \in L^2(G) \rightarrow L^2(G)$ be linear and bounded such that $$ R(T) \subseteq C_b^0(G) = C^0(G) \cap L^{\infty}(G). $$ Then $T$ is a Hilbert-Schmidt operator. If, in addition, $T$ is normal, then there is an orthonormal system $\{\phi_j\}$ of eigenfunctions $T$ so that the corresponding eigenvalues $\lambda_j$ fulfill $$ \lambda_j = {\cal O}(j^{-1/2}). $$