Bergman-Toeplitz operators: Radial component influence

We analyze the influence of the radial component of a symbol to spectral, compactness, and Fredholm properties of Toeplitz operators, acting on the Bergman space. 2 We show that there exist compact Toeplitz operators whose (radial) symbols are unbounded near the unit circle partial derivativeD. Studying this question we give several sufficient, and necessary conditions, as well as the corresponding examples. The essential spectra of Toeplitz operators with pure radial symbols have sufficiently rich structure, and even can be massive The C*-algebras generated by Toeplitz operators with radial symbols are commubative, but the semicommutators (T-a,T-b) = T-a .T-b - T-a.b are not compact in general. Moreover for bounded operators T-a and T-b the operator T-a.b may not be bounded at all.