Abstract. Compared with harmonic Bergman spaces, this paper introduces a new function space which is called the pluriharmonic Hardy space h 2 (T 2 ). We character (semi-) commuting Toeplitz operators on h 2 (T 2 ) with bounded pluriharmonic symbols. Interestingly, these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces, Bergman spaces and harmonic Bergman spaces. Our method for Toeplitz operators on h 2 (T 2 ) gives new insight into the study of commuting Toeplitz operators on harmonic Bergman spaces.
Dual truncated Toeplitz operators on the orthogonal complement of the model space K 2 u (= H 2 ⊖ uH 2 ) with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of K 2 u in L 2 . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator D z , and we even obtain the commutant of all dual truncated Toeplitz operators with bounded analytic symbols. Moreover, we describe the nontrival invariant subspaces of D z .
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