## 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.

#### Rights Information

© 2005 Theta Foundation