Interpolating sequences for \(A^ \infty(\phi)\)-functions (Q1202839)

A positive continuous function \(\varphi\) on \([0,1]\) is said to be normal if there exist constants \(a\) and \(b\), \(0<a<b\), such that \(\varphi(t)/(1- t^ 2)^ a\) is decreasing and \(\varphi(t)/(1-t^ 2)^ b\) is increasing. Let \(A^ \infty(\varphi)\) denote the Banach space of functions \(f\) which are analytic on the unit disc \(D\) and for which \[ \| f\|_{\infty,\varphi}=\sup\{\varphi(| z|)| f(z)|: z\in D\}<\infty. \] Given a sequence \(\{z_ n\}\in D\), define the operator \(T: A^ \infty(\varphi)\to\ell^ \infty\) by \(Tf=\varphi(| z_ n|)f(z_ n)\), \(f\in A^ \infty(\varphi)\). The author proves that \(TA^ \infty(\varphi)=\ell^ \infty\) if and only if the sequence \(\{z_ n\}\) is uniformly separated (i.e., \[ \inf\left\{\prod_{m\neq n}\left|{z_ m-z_ n\over 1-z_ n z_ m}\right|: n=1,2,\dots\right\}>0). \] {}.