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

TL;DR

This paper characterizes semistable unitary operators on L^2(R) that are translation-invariant, symmetric, and locally uniformly continuous under dilation, and describes the associated one-parameter groups of such operators.

Contribution

It provides a new characterization of semistable unitary operators on L^2(R) with specific invariance and continuity properties, and describes the structure of related one-parameter groups.

Findings

Characterization of semistable unitary operators under specified conditions

Description of one-parameter groups of such operators as exponential functions

Identification of operators of the form e^{iβt|d/dx|^α}

Abstract

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 $\alpha,\beta\in\mathbb R$.