Concerning a finite dimensional vector space, we can determine whether two matrices are unitarily equivalent or not by depending on some similarity invariants or Specht's theorem. If we consider an infinite dimensional Hilbert(Banach?) space, what can we use?
I've heard that the position operator and the momentum operator on $L^2(\mathbb{R})$ are unitarily equivalent by the Fourier transformation. However, the way it's shown doesn't seem to be generalizable. For example, how can we classify unitary equivalence classes of multiplication operators? As densely defined operators on $L^2(\mathbb{R})$, $Q_1$ with $Q_1f=xf$ and $Q_2$ with $Q_2f=x^2f$ are unitarily equivalent?