Weak Quasistability and Rajchman measures

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.\

💡 Research Summary

The paper investigates the relationship between weak quasistability of operators and Rajchman measures. After rigorously defining weak stability, weak quasistability, and homogeneous weak quasistability, the author shows that weak quasistability does not guarantee power boundedness. Example 4.1 (the Fögel operator) provides a power‑bounded operator that is weakly unstable yet weakly quasistable, illustrating the separation of these notions. The central result, Theorem 4.3, proves an “either‑or” statement: a weakly quasistable operator on a Banach space must be either power‑bounded or, if it is power‑unbounded, it cannot be coercive (i.e., there is no vector whose orbit norm diverges to infinity). Consequently, coercive power‑unbounded operators cannot be weakly quasistable, and non‑coercive power‑unbounded operators may be weakly quasistable only in a very restricted sense.

The paper then turns to the position operator (U) on (L^{2}(\mathbb{T},\mu)), defined by ((Uf)(z)=zf(z)). Proposition 5.3 establishes a precise equivalence: (U) is weakly stable if and only if the finite measure (\mu) on the unit circle is a Rajchman measure (its Fourier coefficients tend to zero). This recovers the classical characterization of Rajchman measures via operator stability.

The most novel contribution is Theorem 7.3, which shows that for any finite continuous measure (\mu) on the unit circle, the position operator is always weakly quasistable, regardless of whether (\mu) is Rajchman. The proof constructs sparse subsequences of Fourier indices along which the inner products (\langle U^{n_j}f,g\rangle) tend to zero, thereby satisfying the weakened convergence requirement of weak quasistability. This demonstrates that the Rajchman condition is necessary for weak stability but not for weak quasistability.

Further, Theorem 6.1 discusses orthonormal bases ({z_k}) in (L^{2}(\mathbb{T},\mu)) and shows that even when (\mu) is Rajchman, distinct basis elements need not be orthogonal, highlighting subtle spectral properties of Rajchman measures.

Overall, the work clarifies the hierarchy among strong stability, weak stability, and weak quasistability, links these concepts to spectral properties (especially the absence of unit‑circle spectrum for weakly quasistable operators), and provides a nuanced picture of how Rajchman measures govern the behavior of the position operator. The introduction of the “coercive” versus “non‑coercive” dichotomy for power‑unbounded operators is a valuable addition that may inspire further research on operator dynamics and harmonic analysis.