On lower eigenvalue bounds for Toeplitz operators with radial symbols in Bergman spaces

We consider Toeplitz operators in different Bergman type spaces, having radial symbols with variable sign. We show that if the symbol has compact support or decays rapidly, the eigenvalues of such operators cannot decay too fast, essentially faster than for a sign-definite symbol with the same kind. On the other hand, if the symbol decays not sufficiently rapidly, the eigenvalues of the corresponding operator may decay faster than for the operator corresponding to the absolute value of the symbol