Let $X\subset \mathbb{C}^d$ be a domain.
A function (kernel) $K:X\times X\to \mathbb{C}$ is called sesqui-holomorphic if it is holomorphic in the first variable, and anti-holomorphic in the second variable
A kernel $K:X\times X\to \mathbb{C}$ is called positive (semi-)definite if for every $x_1,...,x_n\in X$ the matrix $[K(x_i,x_j)]_{i,j=1}^n$ is positive (semi-)definite. Note that this condition implies that $K$ is Hermitean, i.e. $K(y,x)=\overline{K(x,y)}$.
For a kernel $K:X\times X\to \mathbb{C}$ define it's diagonal $\widehat{K}:X\to\mathbb{C}$ by $\widehat{K}(x)=K(x,x)$. It is known that if for sesqui-holomorphic kernels $K$ and $L$ we have $\widehat{K}=\widehat{L}$, then in fact $K=L$ (this follows from Theorem 7, section II.4 in the book Bochner, Martin - Several Complex Variables). I am interested in a strengthening of that fact:
Let $\{K_n\}_{n\in\mathbb{N}}$ and $K$ be sesqui-holomorphic Hermitean (positive definite if it helps) kernels on $X$ such that $\widehat{K_n}\to\widehat{K}$ uniformly on compacts sets in $X$. Does it follow that $K_n\to K$ uniformly on compacts sets in $X\times X$?
