We study the skew-commutativity of Toeplitz operators and dual Toeplitz operators on the Bergman space of the unit disk. For bounded harmonic symbols, we characterize when two Toeplitz operators are skew-commuting. For general bounded measurable symbols, we give necessary and sufficient conditions for the skew-commutativity of dual Toeplitz operators. Our results characterize skew-commutativity in both settings and show that the conditions are more restrictive than in the commutative case.

We characterize the boundedness and compactness of dual Toeplitz operators on the orthogonal complement of the generalized Fock space. We study the problem when the finite sum of the dual Toeplitz products is compact. Additionally, we also consider when the sum of the dual Toeplitz operators is equal to another dual Toeplitz operator.

We give some characterizations of dual Toeplitz operators acting on the orthogonal complement of the Bergman space over bounded symmetric domains. Our main result characterizes those finite sums of products of Toeplitz operators that are themselves dual Toeplitz operators. Furthermore, we obtain a necessary condition for such finite sums of dual Toeplitz products to be compact. As an application of our main result, we derive a sufficient and necessary condition for when the (semi-)commutators of dual Toeplitz operators is zero. Notably, we find that a dual Toeplitz operator is compact if and only if it is the zero operator.