Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is:
$$
K (t,s) = \sum\_{i=1}^\infty \lambda\_i \phi\_i (t) \phi\_i (s)
$$
with $\{ \phi\_i (t): i \in \mathbb{N}\}$ orthonormal and $\lambda\_1 > \lambda\_2 > \cdots > 0$. Then define
$$ 
K^{1/2} (t,s) = \sum\_{i=1}^\infty \sqrt{\lambda\_i} \phi\_i (t) \phi\_i (s) .
$$
$K(t,s)$ is continuous on $[0,1] \times [0,1]$, is $K^{1/2} (t,s)$ also continuous? Furthermore, if $K \in C^{k}([0,1]^2)$ for some known $k \geq 1$, can we say something about $K^{1/2}$, for example, does $K^{1/2} \in C^{r}([0,1]^2)$ for some $r$ related to $k$?



Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?
**No**.


For example, the Green function for $-\Delta$ (the 1-D Laplace operator) in the interval $(0, 1)$ with Dirichlet boundary condition is continuous (it is $\min(x (1 - y), (1 - x) y)$), but the Green function for the square root of this operator is unbounded: it has logarithmic singularity near the diagonal. This follows, for example, from formula (2.4) in [this paper](https://link.springer.com/content/pdf/10.1007%2Fs00440-002-0251-1.pdf) by R. Song and Z. Vondraček. (For further information, see [here](https://arxiv.org/abs/1610.00872)).


