The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \delta_{i, k}$$
For $i=k$, these conditions in particular say that rows of the matrix $[x_{i,j}(p)]$ has unit $l_{p+2}$ norm.
We are able to establish the finiteness of the solution for fixed $p> 0$ in the following article https://arxiv.org/pdf/2302.00664. We are further trying to see what we can say about the behaviour of the entries $x_{i,j}(p)$ as $p$ varies. For example, we expect that for $p>0$,they should be monotone. This has been observed in dimension $3$ explicitly, and also supported by the intuition that the unit $l_p$ ball inflates in a particular fashion as $p$ increases. But we are not able reason it. We failed to also trace any discussion of a similar system of equations in the literature. Can someone please shed light onto the problem or suggest some reference.
