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

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 36-53

Résumé

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 $[0,a)$, $0$. 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.