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

Abstract

A general scheme which enables us to consider convolution operators with measures acting in a wide class of spaces of distributions on the interval [0, a), 0<a<∞, is represented. It is proved that if a measure μ is a weak generator of the algebra of measures on [0, a), then C_μ (the convolution operator with μ) is unicellular. We give a condition for a measure μ under which the unicellularity of C_μ implies that μ is a weak generator of the algebra of measures. The following statement is also proved. Let$\theta (z) = e^{ - a\frac{{1 + z}}{{1 - z}}}$, K_θ=H²⊖θH², and let P_θ be the orthogonal projector from H² onto K_θ; in addition, let μ be a weak generator of the algebra of measures on [0, a) and$\varphi (z) = (F^{ - 1} \mu )(i\tfrac{{z + 1}}{{z - 1}})$, z ∈$\mathbb{D}$ (here$\mathbb{D}$ is the unit disk and F^-1 is the inverse Fourier transformation). Let ψ∈H^∞ and let p be a polynomial such that p o(ψ−φ)∈θH^∞. Then the operator x→P_θψx, acting in K_θ, is unicellular. Bibliography: 13 titles.