No entire inner functions (Q2300130)

It is well known that the only entire inner functions in the classical Hardy space \(H^2(\mathbb{D})\) are normalized monomials. The authors discuss the similar problem for other reproducing kernel Hilbert spaces of analytic functions on the unit disk. In particular, they consider the weighted Hardy spaces \(H^2_\omega(\mathbb{D})\) with \(\omega=(\omega_n)_{n\ge0}\), \(\omega_0=1\), and \(\omega_n>0\), \(n=1,2,\ldots\), satisfying \[ \lim_{n\to\infty} \frac{\omega_n}{\omega_{n+1}}=1, \qquad \sup_{n\le k\le 2n}\omega_k\le C\omega_n, \ \ n=1,2,\ldots, \] and the standard inner product \(\langle,\rangle_\omega\) with weights \(\omega_n\). A function \(f\in H^2_\omega(\mathbb{D})\) is called inner, if \(\langle z^n f, f\rangle_\omega=\delta_{n,0}\) for all positive integers \(n\).\par A Shapiro-Shields function in \(H^2_\omega(\mathbb{D})\) is a function which is a proper analogue of the finite Blaschke products in \(H^2(\mathbb{D})\). The space \(H^2_\omega(\mathbb{D})\) has no extraneous zeros if all zeros of all Shapiro-Shields functions in the space are regular. The main result of the paper claims that the only entire inner functions for such weighted Hardy spaces are normalized monomials.