Interpolation problems on the spectrum of \(H^{\infty}\)
Let \(H^{\infty}\) be the uniform algebra of all bounded holomorphic functions in the open unit disk \(\mathbb D\). Its spectrum, denoted by \(M(H^{\infty})\), is the set of all (nonzero) multiplicative linear functionals on \(H^{\infty}\) endowed with the weak-\(\ast\)-topology. The pseudohyperbolic distance between two points \(x\) and \(y\) in \(M(H^{\infty})\) is given by \(\rho(x,y)=\sup\{|f(x)|:\,\| f\|_{\infty}\leq1,\,f(y)=0\}\). The Gleason part associated with the point \(x_n\) is \(P(x_n)=\{y\in M(H^{\infty}): \rho(y,x_n)<1\}\). A Blaschke product is called thin if its zeros inside \(\mathbb D\) form a thin sequence, that is, \(\lim_{j\to\infty}\prod_{n\neq j}\rho(z_n,z_j)=1\). In the paper under review, the author characterizes those sequences in \(M(H^{\infty})\) whose Nevanlinna-Pick interpolation problems admit thin Blaschke products as solutions. Further, it is studied under which conditions there is a Blaschke product \(B\) with prescribed zero-set distribution and solving problems of the form \(B(x)=f_n(x)\) for every \(x\in P(x_n)\), where \(P(x_n)\) is the Gleason part associated with the point \(x_n\) and where \((f_n)\) is an arbitrary sequence of functions in the unit ball of \(H^{\infty}\). Using the above result, a new characterization of Carleson-Newman Blaschke products is obtained in terms of bounded universal functions, a result first proved by \textit{E.\,Gallardo-Gutierrez} and \textit{P.\,Gorkin} [Oral communication of the statement at the MFO in Oberwolfach, July (2006)].
scientific article; zbMATH DE number 5602532
scientific article