I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting contractions and the corresponding von Neumann inequality, and so-called pairs of $q$-commuting contractions), but I still fail to see what the current open problems in dilation theory are.
Could anyone maybe point them out? I know that a general problem is e.g. for a complex polynomial $p$, finding such a value $C_n$ that $$p(T_1, \ldots, T_n) \le C_n \cdot \sup_{z \in \overline{\mathbb{D}^n}} | p(z)|$$ and that there are some partial results. But this is not what I'm asking about as I am familiar with the general problems in dilation theory.
What I'm asking is what are some specific problems in dilation theory, which yet have to be solved by someone. I ask about specific problems, because I feel as if just knowing the general problems is getting me nowhere and I feel as if I'd need some specific problem to work on.
What I mean by "specific" problem? An open question which has been left to others after a mathematician has worked on a specific case/example in dilation theory (in the realms of the "general problem"). An (already solved) example of a specific problem would be finding out whether an $n$-tuple of $3 \times 3$ matrices satisfies the von Neumann inequality after having it proved for $2 \times 2$ (the answer is yes!). For the curious out there: It does not work for $4 \times 4$ matrices as there exists a counterexample.
I'm looking forward for your answers, it will be a great help for my PhD studies.
Thanks