Toeplitz operators on generalized Bergman-Hardy spaces - Lusky, Wolfgang (Mathematica Scandinavica, 2001)
We study the Toeplitz operators T_f: H_2 → H_2, for f L_∞, on a class of spaces H_2 which in- cludes, among many other examples, the Hardy and Bergman spaces as well as the Fock space. We investigate the space X of those elements f L_∞ with _j \|T_f-T_f_j\|=0 where (f_j) is a sequence of vector-valued trigonometric polynomials whose coefficients are radial functions. For these T_f we obtain explicit descriptions of their essential spectra. Moreover, we show that f X, whenever T_f is compact, and characterize these functions in a simple and straightforward way. Finally, we determine those f L_∞ where T_f is a Hilbert-Schmidt operator.