Range Of Toeplitz Operator

In the first part of this paper, we got some sufficient and necessary conditions about surjective Toeplitz operators, through investigating symbol of Toeplitz operator via J.Bourgain fractorization theorem. For a single Toeplitz operator, it was obtained that whether Toeplitz operator was surjective merely had something with its unimodular symbol. We detailed Toeplitz operator with unomodular symbol, and generated the theorem proved by Michael Sand in [2] from H∞ to H∞ + C(T). In the second part of this paper, the author discussed simply when multipliction of two Toeplitz operators with symbol in H∞+ C(T) or in L∞ .We got two conclusions by analyzing symbol of Toeplitz operators. The last, we investigated whether multiplication of two Toeplitz operators, which range included all non-cyclic vectors of backward shift, is surjective , and partly answered the question about surjective Toeplitz algebra posed by Michael Sand in[2].