The distance to the analytic Toeplitz operators (Q1080140)

Let \({\mathcal T}(H^{\infty})\) denote the algebra of analytic Toeplitz operators. It is reflexive and maximal abelian, so there are three natural measures of the distance to \({\mathcal T}(H^{\infty})\). They are (i) \(d(A)=\inf \{\| A-T_ h\|:h\in H^{\infty}\},\) (ii) \(\delta (A)=\sup \{\| AT_ h-T_ hA\|:h\in H^{\infty}\), \(\| h\| \leq 1\}\), and (iii) \(\beta (A)=\sup \{\| P^{\perp}_{\Omega}AP_{\omega}\|:P_{\Omega}=T^*_{\omega}T_ {\omega}\), \(\omega\) inner\(\}\). This latter estimate is based on Beurling's theorem that \(\omega H^ 2\) are the invariant subspaces of \({\mathcal T}(H^{\infty})\) and Sarason's observation that it is a reflexive algebra. The main result is that these quantities are comparable: \[ (1/2)d(A)\leq \delta (A)\leq 2d(A)\quad and\quad \beta (A)\leq d(A)\leq 19\beta (A) \] for all operators A. As a corollary, one obtains that if \({\mathcal T}\) is any unitial, weak \({}^*\) closed subalgebra of Toeplitz operators, then \[ dist(A,{\mathcal T})\leq 39 \sup \{\| P^{\perp}TP\|:P\in Lat {\mathcal T}\} \] for all A in \({\mathcal B}(H)\).