Univalent functions in cyclic vectors in \(Q_p\) space
Let \(H(\mathbb D)\) be the set of holomorphic functions on the open unit disk \(\mathbb D\) and \(X\subset H(\mathbb D)\) be a Banach space such that the polynomials are norm (or weak\(^*\)) dense in \(X\). A cyclic vector in \(X\) is a function \(f\), the products of which with polynomials are dense in \(X\). The space \(Q_p\), \( 0\leq p<\infty\), consists of functions \(f\in H(\mathbb D)\) such that \[ \|f\|_{Q_p}=\sup_{a\in{\mathbf D}}\bigg(\int_{\mathbf D}|f'(z)|^2\bigg(1-\bigg|\frac{a-z}{1-{\overline a}z}\bigg|^2\bigg)^p \mathrm{d}m(z)\bigg)^{1/2}<\infty, \] where \(m\) is the Lebesgue area measure on \(\mathbb D\), equipped with the norm \(\|f\|_*=|f(0)|+\|f\|_{Q_p}\). Note that \(Q_0\) is the Dirichlet space, \(Q_1=\mathrm{BMOA}\) and the \(Q_p\) are Bloch spaces when \(1<p<\infty\). The space \(Q_{p,0}\subset Q_p\) consists of the functions \(f\in Q_p\) such that \[ \lim_{|a|\to 1}\bigg(\int_{\mathbf D}|f'(z)|^2\bigg(1-\bigg|\frac{a-z}{1-{\overline a}z}\bigg|^2\bigg)^p \mathrm{d}m(z)\bigg)^{1/2}=0. \] The authors show that a univalent function \(f\) in \(Q_p\) (resp. \(Q_{p,0}\)) is cyclic in \(Q_p\) (resp. \(Q_{p,0}\)) if and only if it does not vanish on \(\mathbb D\).
scientific article; zbMATH DE number 6305066
scientific article