Abstract

Given a Hilbert space $X$ consisting of functions that are analytic in the unit disk or unit polydisk, a standard problem is to classify the invariant subspaces with respect to the operator induced by multiplication by the coordinate function(s). As a first step, one often tries to find the cyclic vectors $f\in X$. Phrased differently, $f\ in X$ is cyclic if there exists a sequence of polynomials (in one or two variables) such that $\|p_nf-1\|_X\to 0$ as $n\to \infty$.

In recent joint work with B\’en\’eteau, Condori, Liaw, and Seco, we have have studied this problem in Dirichlet-type spaces: for certain subclasses of functions, we determine explicitly the best approximants $(p_n)$, and obtain sharp rates of decay for the associated norms.

Apart from basic complex analysis (power series, Cauchy’s formula) and functional analysis (Hilbert space theory), no specialized background material will be assumed for this talk, and the results will be illustrated with examples.