Commutators and Semi-commutators of Monomial Toeplitz Operators on the Pluriharmonic Hardy Space
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Commutators and Semi-commutators of Monomial Toeplitz Operators on the Pluriharmonic Hardy Space∗ Yingying ZHANG1
Xingtang DONG1
Abstract In this paper, the authors completely characterize the finite rank commutator and semi-commutator of two monomial Toeplitz operators on the pluriharmonic Hardy space of the torus or the unit sphere. As a consequence, many non-trivial examples of (semi-)commuting Toeplitz operators on the pluriharmonic Hardy spaces are given. Keywords Toeplitz operator, Pluriharmonic Hardy space, Commutator, Semicommutator, Monomial symbol 2000 MR Subject Classification 47B35, 47B47
1 Introduction Let T be the unit circle in the complex plane C and the torus Tn be the Cartesian product of n copies of T. Let dµ be the normalized Haar measure on Tn . The Hardy space H 2 (Tn ) is the closure of the analytic polynomials in L2 (Tn , dµ) (or L2 (Tn )). It is well known that H 2 (T) + H 2 (T) ∼ = L2 (T). However, for n ≥ 2, H 2 (Tn ) + H 2 (Tn ) $ L2 (Tn ). So we shall suppose n ≥ 2 to avoid trivialities throughout the paper and define the pluriharmonic Hardy space h2 (Tn ) by h2 (Tn ) = H 2 (Tn ) + H 2 (Tn ). See [3] for more information about the pluriharmonic Hardy space h2 (Tn ). Similarly, let dσ be the surface area measure on the unit sphere Sn , the pluriharmonic Hardy space h2 (Sn ) denotes the closed subspaces of all pluriharmonic functions in L2 (Sn , dσ) (or L2 (Sn )). Let Q be the orthogonal projection from L2 (Ωn ) onto h2 (Ωn ), where Ωn denotes Tn or Sn . The Toeplitz operator with symbol f in L∞ (Ωn ) is defined by Tf (h) = Q(f h) for functions h ∈ h2 (Ωn ). It is safe to use the same notation Tf to denote the Toeplitz operators on both h2 (Tn ) and h2 (Sn ), as we will always specify the space on which the operator Tf acts. For two Toeplitz operators Tf1 and Tf2 on h2 (Ωn ), we define their commutator and the semi-commutator by [Tf1 , Tf2 ] = Tf1 Tf2 − Tf2 Tf1 and (Tf1 , Tf2 ] = Tf1 Tf2 − Tf1 f2 , respectively. On the Hardy space H 2 (T), Brown and Halmos [2] first obtained a complete description of bounded symbols of (semi-)commuting Toeplitz operators. Later, some related problems Manuscript received April 3, 2018. of Mathematics, Tianjin University, Tianjin 300354, China. E-mail: [email protected] [email protected] ∗ This work was supported by the National Natural Science Foundation of China (Nos. 11201331, 11771323). 1 School
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were studied by many authors (see [7] and references there). However, the function theory on Ωn is quite different from and much less understood than that on T. For example, the complete characterization of (semi-)commuting Toeplitz operators on the Hardy space H 2 (Tn ) were obtained only when n = 2 (see [4, 8]). Zheng [11] characterized commuting Toeplitz operators with bounded pluriharmonic symbols on H 2 (Sn ). In the setting of pluriharmonic Hardy spaces, Liu and Ding [9] obtained a characterization of (sem
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