This paper provides a class of complex symmetric weighted composition operators on H 2 (D) to includes the unitary subclass, the Hermitian subclass and the normal subclass obtained by Bourdon and Noor. A characterization of algebraic weighted composition operator with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not necessarily linear fractional.2010 Mathematics Subject Classification. 47B33, 47B38.

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In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomialtype Toeplitz operators. Our results are new even for the unit ball.2010 Mathematics Subject Classification. Primary 47B35; Secondary 32A36.