Sums of dual Toeplitz products on the orthogonal complements of Fock-Sobolev spaces

We consider dual Toeplitz operators on the orthogonal complements of the Fock-Sobolev spaces of all nonnegative real orders. First, for symbols in a certain class containing all bounded functions, we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero. Next, for bounded symbols, we construct a symbol map and exhibit a short exact sequence associated with the C^*-algebra generated by all dual Toeplitz operators with bounded symbols.