Pomona Faculty Publications and Research

Title

Conjugation, the Backward Shift, and Toeplitz Kernels

Article

Department

Mathematics (Pomona)

2005

Keywords

conjugation, backward shift, Toeplitz kernels, pseudocontinuations, Darlington synthesis problem, electrical network theory

Abstract

For each outer function $\Omega$ in the Smirnov class and each $p \in (0,\infty)$, we define a subspace $\mathcal{N}_{\Omega}^p$ of $H^p$ that carries an operation analogous to complex conjugation. Using these subspaces, we explicitly describe the invariant subspaces and noncyclic functions for the backward shift operator on $H^p$ for $p \in [1,\infty)$ and $p \in (0,\infty)$, respectively. We also discuss pseudocontinuations, the Darlington synthesis problem from electrical network theory, and the kernels of Toeplitz operators.