\(dist\)-formulas and Toeplitz operators (Q2346125)

Let \(H^2=H^2(\mathbf T)\) be the Hardy space over the unit circle \(\mathbf T\) and let \(P\) be the Hilbert space orthogonal projection of \(L^2=L^2(\mathbf T)\) onto \(H^2\). Given a nonconstant function \(\varphi\in L^\infty(\mathbf T)\), let \(H_\varphi: H^2 \to L^2 \ominus H^2\) be the Hankel operator defined by \(H_\varphi f = \varphi f - P(\varphi f)\) for \(f\in H^2\) and denote by \(\operatorname{dist}(\varphi,\mathcal F_{\mathrm{const}})\) the distance from \(\varphi\) to the set \(\mathcal F_{\mathrm{const}}\subset L^\infty\) of all constant functions on \(\mathbf T\). In this paper, the authors obtain the following distance estimate: \[ \max\left\{ \sup_{\theta\in (\Sigma)}\| H_{\overline\varphi}^\ast H_{\overline\theta}\|, \| H_\varphi\|\right\}\leq \operatorname{dist}(\varphi,\mathcal F_{\mathrm{const}})\leq \|\varphi\|_{L^\infty}, \] where \((\Sigma)\) denotes the set of all inner functions. Other related distance estimates are also obtained.