Abstract. We study Toeplitz operators on the harmonic Bergman spaces on bounded smooth domains. Two classes of symbols are considered; one is the class of positive symbols and the other is the class of uniformly continuous symbols. For positive symbols, boundedness, compactness, and membership in the Schatten classes are characterized. For uniformly continuous symbols, the essential spectra are described.
Abstract. Motivated by a recent work of Loaiza et al. for the holomorphic case on the disk, we introduce and study the notion of Schatten-Herz type Toeplitz operators acting on the harmonic Bergman space of the ball. We obtain characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions. Our characterization in terms of Berezin transforms settles a question posed by Loaiza et al.
On the setting of bounded smooth domains, we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman space into another in terms of certain Carleson and vanishing Carleson measures.
Recently, Schatten-Herz type Toeplitz operators have been studied on the Bergman spaces and the harmonic Bergman spaces. Motivated these results, we study characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman spaces.
We study characterizations of arbitrary positive Toeplitz operators of Schatten (or Schatten-Herz) type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman space. (2000). Primary 47B35; Secondary 31B05.
Mathematics Subject Classification
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