Indices, characteristic numbers and essential commutants of Toeplitz operators

Abstract

For an essentially normal operator T, it is shown that there exists a unilateral shift of multiplicity m in C^*(T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as C^*-algebras. Finally, we construct a natural C^*-algebra ε + ε_* on the Bergman space L ${}_{a}^{2}$ (B_n), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.