A note on semistable unitary operators on $L^2(\mathbb{R})$

In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under dilation. As a consequence, we characterize one-parameter groups formed by such operators, which are of the form $e^{i\beta t|{d}/{dx}|^\alpha}$, with $\alpha,\beta\in\mathbb R$.