Invariant subspaces of the Dirichlet shift and pseudocontinuations

Abstract:

In this paper we study extremal functions for invariant subspaces $\mathcal {M}$ of the Dirichlet shift, i.e., solutions $\varphi$ of the extremal problem \[ \sup \{ |{f^{(n)}}(0)|/{\left \| f \right \|_D}:f \in \mathcal {M},f \ne 0\} \]. Here $n$ is the smallest nonnegative integer such that the sup is positive. It is known that such a function $\varphi$ generates $\mathcal {M}$. We show that the derivative $(z\varphi )\prime$ has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where $\varphi$ is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.