The Bergman shift operator on polynomial lemniscates (Q2257721)

Let \(\mu\) be a Borel measure with support given by an infinite compact subset of the plane \(\mathbb{C}\). Let \(L^2(\mathbb{C};\mu)\) denote the space of square integrable functions relative to the measure \(\mu\). Performing Gram-Schmidt orthogonalization to the set of functions \(\{1,z,z^2,\ldots\}\) one arrives at a collection of orthogonal polynomials \(\{\varphi_n(z;\mu)\}\). Let \(\kappa_n=\kappa_n(\mu)\) denote the leading coefficient of the polynomial \(\varphi_{n}(z;\mu)\). Any weak limit of the sequence of measures \(\{\left| \varphi_{n}(z;\mu)\right|^2d\mu(z)\}\) is called a \textit{weak asymptotic measure}. The Bergman shift operator acts on \(L^2(\mathbb{C};\mu)\) by \(\mathcal{M}f(z)=zf(z)\) (pointwise multiplication by \(z\)). Let \(\mathcal{P}\) denote the closure of the span of polynomials inside \(L^2(\mathbb{C};\mu)\). Let \(M\) denote the matrix representation of the map \(\mathcal{M}\) acting on \(\mathcal{P}\). Let \(\mathcal{R}\) denote the right shift operator on \(\ell^2(\mathbb{N})\). For a measure \(\mu\) as above and a monic polynomial \(P(z)\) of degree \(q\geq 1\) and \(r>0\), the main result of this paper is that \(\{(P(M)-r\mathcal{R}^q)\mathcal{R}^n\}\) converges strongly to \(0\) if and only if \(\lim_{n} \kappa_n \kappa_{n+q}^{-1}=r\) and every weak asymptotic measure is supported on \(\{z: \left| P(z)\right|=r\}\). Numerous corollaries of the result are obtained which provide additional information about the properties of the orthogonal polynomials \(\{\varphi_n(z;\mu)\}\).