Interpolating sequences for weighted spaces of analytic functions on the unit ball of a Hilbert space
Given a complex Hilbert space $E$ with the unit ball $B_E$, and a positive, continuous function $v$ on $B_E$ (weight), the authors study the class $H^\infty_v(E)$ of scalar, analytic functions $f$ on $B_E$ so that \[ \|f\|_v:=\sup_{x\in B_E} v(x)|f(x)|<\infty. \] A sequence $(w_n)\in B_E$ is said to be interpolating for $H^\infty_v(E)$ if the map \[ S:\ H^\infty_v(E)\to\ell^\infty, \quad S(f)=(v(w_n)f(w_n))_{n\ge1}, \] is onto. In case $S$ has a linear right inverse, $(w_n)$ is a linear interpolating sequence. \par Let $v_\alpha(x):=(1-\|x\|^2)^{\alpha}$, $\alpha\ge0$. The authors show that each interpolating sequence for $H^\infty_{v_\alpha}(E)$ is hyperbolically separated. Conversely, if $(w_n)$ is hyperbolically $R$-separated for some $R > 0$, then it is a linear interpolating sequence for $H^\infty_{v_\alpha}(E)$ (Theorem 5.7). \par Similar results hold for some general weights as well as for Bloch-type spaces.
scientific article; zbMATH DE number 7037769
scientific article