Uncertainty principles revisited
One form of the uncertainty principle states that for (densely defined) self-adjoint operators \(A,B\) on a Hilbert space \({\mathcal H}\) one has \[ \| (A-a) u\| \| (B-b)u\| \geq \frac{1}{2} | \langle (AB-BA)u,u\rangle | \] whenever \(u\in \text{ Dom}\, (AB) \cap \text{ Dom}\, (BA)\) and \(a,b\in \mathbb R\), while equality holds precisely when \((A-a)u\) and \((B-b)u\) differ by a purely imaginary scalar multiple (cf., e.g., \textit{G.B. Folland} [``Harmonic analysis in phase space'' (Annals of Mathematics Studies 122, Princeton University Press) (1989; Zbl 0682.43001)]). As the author points out, not every operator of interest is self-adjoint. Thus the author extends this uncertainty principle to the case in which \(A,B\) are symmetric (i.e., \(\langle Af, g\rangle = \langle f, Ag\rangle \) whenever both \(f,g\in \text{ Dom}(A)\subset \text{ Dom}\, (A^\ast)\)) or normal (i.e., closed, densely defined and \(AA^\ast = A^\ast A\)). The cases that give rise to equality are also identified. The author illustrates that this generalization leads to a simple proof of a certain inequality for the so-called Dunkl operator on a weighted \(L^2\)-space on \([0,1]\) as well as a new uncertainty principle for natural versions of the position and momentum operators on \(L^2\) of the sphere in \(\mathbb R^3\).
scientific article; zbMATH DE number 1893817