A note on the spectral area of Toeplitz operators (Q2796722)

The main result of the paper (Theorem 2.1) states the following: Let \(\varphi \in L^{\infty}(\partial \mathbb{D})\) and \(\varphi = f + \overline{T_{\overline{h}}f}\), for \(f,\, h \in H^{\infty}\), \(\| h\| _{\infty} \leq 1\), and \(h(0) = 0\). Then NEWLINE\[NEWLINE \| [T^*_{\varphi},t_{\varphi}]\| \geq \int |f - f(0)|^2\frac{d\theta}{2\pi} = \| P(\varphi) - \varphi(0)\| ^2_2, NEWLINE\]NEWLINE where \(P\) is the orthogonal projection of \(L^{\infty}(\partial \mathbb{D})\) onto \(H^{\infty}\). NEWLINENEWLINETogether with the celebrated Putnam inequality for hyponormal operators this leads to the following corollary. Under the above assumptions, \(\mathrm{Area\,}\big(\mathrm{sp\,}(T_{\varphi})\big) \geq \pi \| P(\varphi) - \varphi(0)\| ^2_2\).