Some results for Toeplitz operators on the Bergman space

We consider the Toeplitz operators on the Bergman space La2(D) on the unit disc DD. In terms of reproducing kernels the invariant subspaces of Bergman space Toeplitz operator is studied. We also investigate invertibility of Toeplitz operators in terms of Berezin symbols. Moreover, the Hankel operators and Berezin symbols are used to characterize compactness of operators 2Tφψ − TφTψ − TψTφ for bounded harmonic functions φ and ψ.