Kernel of Continuous Functional Calculus of Integral Operator

Question

Suppose we have a symmetric bounded function $k:[0,1] \times [0,1] \rightarrow \mathbb{R}$ that induces the integral operator $T_k: L^2([0,1]) \rightarrow L^2([0,1])$, $$ (T_k f)(u) = \int_0^1 k(u,v) f(v) dv, $$ for $f \in L^2([0,1])$. Let $h: \mathbb{R} \rightarrow \mathbb{R}$ be (at least) continuous. Is there a closed form for applying the continuous functional calculus to integral operators of this form? Under which conditions on $h$ do we know that $h(T_k)$ is again an integral operator? And if we know that $h(T_k) = T_{h^k}$ for some symmetric bounded function $h_k:[0,1] \times [0,1] \rightarrow \mathbb{R}$, is there a way to calculate $h_k$ explicitely? I mention that I do not assume that we know the eigendecomposition of $T_k$.