A Bergman space HL2(Bd, du_eta) is non-zero if and only if eta>-1. We define new spaces,HL2(Bd,eta), which are the same as Bergman spaces if eta>-1 but non-zero for eta>-(d+1). We define Toeplitz operators associated with polynomials to the spaces HL2(Bd,eta) when -(d+1)<eta =< -1. We can show that some properties of Toeplitz operators do not hold in this range. We also use some properties of Berezin transforms and Hilbert-Schmidt operators to extend the definition of Toeplitz operators associated with L2(Bd, d au), where d au is the hyperbolic volume measure, to the spaces HL2(Bd,eta) when -(d+2)/2<eta=< -1.