A note on cyclic vectors in \(\mathcal{Q}_p\) space (Q401499)

For \(0< p<1\), let \(Q_p\) be the space of all holomorphic functions on the disk \(D\) satisfying \(\sup_{a\in D} I_a(f)<\infty\), where NEWLINE\[NEWLINEI_a(f):=\int_D |f'(z)|^2 (1-|\rho_a(z)|^2)^p d\sigma_2(z)<\infty,NEWLINE\]NEWLINE and \(Q_{p,0}=\{f\in Q_p: \lim_{|a|\to 1} I_a(f)=0\}\). It is shown that if \(f\) is invertible in \(Q_p\), then \(f\) is weak*-cyclic in \(Q_p\) and if \(f\) is invertible in \(Q_{p,0}\), then \(f\) is norm-cyclic in \(Q_{p,0}\).