A generalization of Toeplitz operators on the Bergman space

Abstract

If μ μ is a finite measure on the unit disc and k ⩾ 0 k⩾0 is an integer, we study a generalization derived from Engli\v{s}'s work, T ( k ) μ Tμ(k), of the traditional Toeplitz operators on the Bergman space \berg \berg, which are the case k = 0 k=0. Among other things, we prove that when μ ⩾ 0 μ⩾0, these operators are bounded if and only if μ μ is a Carleson measure, they are compact if and only if μ μ is a vanishing Carleson measure, and we obtain some estimates for their norms. Also, we use these operators to characterize the closure of the image of the Berezin transform applied to the whole operator algebra.