Weak Quasistability and Rajchman measures

TL;DR

This work investigates how weak stability and weak quasistability of operators interact with Rajchman measures in the setting of multiplication/position operators on ${L^2({\mathbb T},\mu)}$. It establishes that weakly quasistable operators are constrained to be either power bounded or noncoercively power unbounded, and proves that for the position operator, Rajchman-ness of the underlying measure corresponds to weak stability only when ${\{z^k\}}$ forms an orthonormal basis; in general, Rajchman measures do not guarantee such a basis. It further shows that the position operator is weakly quasistable for every finite continuous measure, and develops a spectrum of equivalences and counterexamples linking Rajchman properties with weak stability and orthogonality of trigonometric systems. These results illuminate the delicate boundary between stability notions and Fourier-analytic properties of measures on the unit circle, with implications for spectral theory of unitary operators and basis questions in ${L^2({\mathbb T},\mu)}$.

Abstract

It is shown that weak quasistability does not imply power boundedness, but coercive power unbounded operators cannot be weakly quasistable.\ Although a finite measure over the unit disc is a Rajchman measure if and only if the position operator is weakly stable, it is shown that the position operator is weakly quasistable for every finite continuous measure over the unit disc.\ Corollaries linking Rajchman measures with weak stability and weak quasistability follow the above results.\