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# 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.

Keywords:  Invariant subspaces, backward shift operator, Toeplitz operators, pseudocontinuation, Smirnov class, unitary matrices, Darlington synthesis, cyclic functions, noncyclic functions, inner functions.