Essential commutant of analytic Toeplitz operators
Let \(S_n\) be the unit sphere in \(\mathbb{C}^n\) and for \(f\in L^\infty (S_n)\), let \(T_f,H_f\) be the corresponding Toeplitz and Hankel operator, respectively, on the Hardy space \(H^2(S_n)\). Denote \(\mathcal A=\{f\in L^\infty(S_n): H_f\) is compact\} and \(\mathcal A _c =\{ f\in L^\infty(S_n): T_f T_g -T_g T_f\) is compact \(\forall g\in H^\infty(S_n) \}\). It was shown by \textit{K. R. Davidson} [J. Funct. Anal. 24, 291-302 (1977; Zbl 0343.47022)] that \(\mathcal A_c=\mathcal A=H^\infty+C\) when \(n=1\) and by \textit{A. M. Davie} and \textit{N. P. Jewell} [ibid. 26, 356-368 (1977; Zbl 0374.47011)] that \(\mathcal A_c \supset\mathcal A\supsetneq H^\infty+C\) when \(n>1\). In the paper under review, the authors prove that \(\mathcal A_c =\mathcal A\) holds for all \(n\). The proof uses several ideas from the proof of Davidson for \(n=1\) and some facts about the inner functions on the unit ball of \(\mathbb{C}^n\).
scientific article; zbMATH DE number 1077933