Idempotent, model, and Toeplitz operators attaining their norms

TL;DR

This paper investigates conditions under which certain classes of operators, including idempotent, model, and Toeplitz operators, attain their norm, with a focus on model operators on backward shift invariant subspaces.

Contribution

It proves that the model operator on any backward shift invariant subspace of the Hardy space attains its norm, extending understanding of norm attainment in operator theory.

Findings

Model operators on backward shift invariant subspaces attain their norm.

Characterization of norm-attaining properties for Toeplitz and idempotent operators.

Extension of classical results to a broader class of subspaces.

Abstract

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$ attains its norm. Here $S_{\mathcal{Q}} = P_{\mathcal{Q}}M_z|_{\mathcal{Q}}$, the compression of the shift $M_z$ on the Hardy space $H^2(\mathbb{D})$ to $\mathcal{Q}$.