Universal multipliers for Sub-Hardy Hilbert spaces

To every non-extreme point $b$ of the unit ball of ${\text{hil}}^{\mathrm{\infty }}$ of the unit disk there corresponds a Pythagorean mate, a bounded outer function $a$ satisfying the equation $\mathrm{\mid }a{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2=1$ on the boundary of the disk. We study universal, i.e., simultaneous multipliers for families of de Branges-Rovnyak spaces $\text{hb}$, and develop a general framework for this purpose. Our main results include a new proof of the Davis-McCarthy universal multiplier theorem for the class of all non-extreme spaces $\text{hb}$, a characterization of the Lipschitz classes as the universal multipliers for spaces $\text{hb}$ for which the quotient $b\mathrm{/}a$ is contained in a Hardy space, and a similar characterization of the Gevrey classes as the universal multipliers for spaces $\text{hb}$ for which $b\mathrm{/}a$ is contained in a Privalov class.