Normal Families of Toeplitz Operators on Fractional Parabolic Hardy Spaces

We study Toeplitz operators on fractional parabolic Hardy spaces associated with fundamental solutions of fractional heat operators. For a finite positive Borel measure $\mu$ on the upper half-space, we first recall the characterization of boundedness and compactness of Toeplitz operators in terms of Carleson measures. The main purpose of this paper is to investigate the stability of Toeplitz operators with respect to variations of the fractional parameter $\alpha \in (0,1]$. Assuming that $\mu$ satisfies a uniform Carleson condition on a compact interval of parameters, we prove that the corresponding family of Toeplitz operators depends continuously on $\alpha$ in the operator norm topology. Moreover, this stability property gives rise to a normal family structure of the operator family ${T_\mu^{(\alpha)}}_{\alpha}$, which may be regarded as an operator-theoretic analogue of the classical Montel–Vitali theorem in complex analysis. As immediate consequences, boundedness and compactness are shown to be stable under perturbations of the fractional parameter. This is a revised and extended version of an earlier preprint on stability of Toeplitz operators.