Weak generators of the algebra of measures and unicellularity of convolution operators

A general procedure is constructed, which allows us to consider operators of convolution with measures acting on a large class of spaces of distributions on the segment $[0,a)$, $0<a<\infty$. It is proved that if a measure $\mu$ is a weak generator of the algebra of measures on $[0,a)$, then $C_\mu$ (the operator of convolution with $\mu$) is unicellular. We present a condition on the measure $\mu$ under which unicellularity of $C_\mu$ implies that $\mu$ is a weak generator of the algebra of measures. The following statement is proved as well. Let $\theta(z)=e^{-a\frac{1+z}{1-z}}$, $K_\theta=H^2\ominus\theta H^2$, and let $P_\theta$ be the orthogonal projection from $H^2$ onto $K_\theta$; moreover, let $\mu$ be a weak generator of the algebra of measures on $[0,a)$ and $\varphi(z)=(\mathcal F^{-1}\mu)(i\frac{z+1}{z-1})$, $z\in\mathbb D$ (here $\mathbb D$ is the unit disc, and $\mathcal F^{-1}$ is the inverse Fourier transform). Let $\psi\in H^\infty$ and let $p$ be a polynomial such that $p\circ(\psi-\varphi)\in\theta H^\infty$. Then the operator $x\mapsto P_\theta\psi x$ acting in $K_\theta$ is unicellular. Bibliography: 13 titles.