Indices, characteristic numbers and essential commutants of Toeplitz operators
Let \(\gamma(T)\) denote the characteristic number of an essentially normal operator \(T\) and let \(C^*(T)\) be the \(C^*\)-algebra generated by \(T\). It is shown that there exists a unilateral shift of multiplicity \(m\) in \(C^*(T)\) if and only if \(\gamma(T)|m\). Using this result the author answers (in the negative) a question of \textit{M. Engliš} [Ark. Mat. 30, 227-240 (1992; Zbl 0784.46036)] whether the essential commutants of unilateral and bilateral shifts are isomorphic as \(C^*\) algebras. Let \(B_n\) be the unit ball of \(C^n\). On the Bergman space \(L^2_a(B_n)\) a natural \(C^*\)-algebra is constructed. It is shown that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.
scientific article; zbMATH DE number 1928890