Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 239-250.

Conjugation, the backward shift, and Toeplitz kernels

Authors: Stephan Ramon Garcia
Author institution: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, 93106--3080, USA

Summary: 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.

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