The similarity to normal problem: From Friedrichs' Г-equations to Treil's curvatures

We survey the similarity problem to a normal operator (matrix) both in finite and infinite dimensions. For the matrix case, the problem is to bound the dimensional growth of the "similarity cost constant", which is formally speaking linear but we discuss its possible sublinear growth. For the infinite dimensional case, we are looking for a language to express the similarity phenomenon. In particular, much work is done in terms of the resolvent (growth).