Let $X$ be a compact metric space and $\mu$ be strictly positive Borel measure on $X$. Let $T:L^2(X,\mu)\rightarrow L^2(X,\mu)$ be a self-adjoint, compact, and positive operator on the Hilbert space $L^2(X,\mu)$ with continuous kernel $K:X\times X\rightarrow\mathbb{R}$. In other words, $$Tf(x)=\int_X K(x,y)f(y) d\mu(y)$$ for any $x\in X$ and $f\in L^2(X,\mu)$. My question is the following one.

Q1: It is known that there exist the unique square root operator $T^{1/2}$ of $T$. Then, this $T^{1/2}$ is also an integral operator with some continuous kernel $K':X\times X\rightarrow\mathbb{R}$?

Since I am not an expert of operator theory, maybe this is not research level question. But at least I could not find any literature answer this.

My guess is, probably the answer is false. But I could not come up with any counterexamples.

Q2: If the answer is false, then what is counterexample?

Q3: By adding some mild conditions, are we able to make the statement true?