I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy.
A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, W_2, \ldots , W_n \in M_{s,t}$ such that the following matrix of matrices \begin{align} [(I-W_iW_j^*)k(x_i, x_i)]_{1\le i,j \le m} \end{align} is positive semidefinite.
A kernel is said to have scalar Pick property if it has the $M_{1,1}$ pick property and it is said to have complete Pick property if it has the $M_{s,t}$ pick property for all $s,t \in \mathbb N$.
Is there a class of examples which are scalar Pick but not complete pick? I am aware of McCullough Quiggin characterisation of Complete pick kernel. I tried looking for examples but could not find any.
EDIT: I had deleted this question as Quiggin in his PhD thesis showed a $4\times 4$ matrix which has the scalar Pick property but not the complete Pick property. I still wonder if there is a holomorphic space, which has the scalar Pick property but not the complete Pick property.
UPDATE: Antonio Serra came up a class of examples in his PhD thesis and has published in his paper: https://mathscinet.ams.org/mathscinet-getitem?mr=2187439 .
