Herz Classes and Toeplitz Operators in the Disk

Abstract

In this paper we decompose D into diadic annuli {An : n ∈ N} and consider the class Sp,q of Toeplitz operators Tϕ for which the sequence of Schatten norms Tϕ n p n∈N belongs to q , where ϕn = ϕχA n . We study the boundedness and compactness of the operators in Sp,q and we describe the operators Tϕ, ϕ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols ϕ. (2000). Primary 47B35, Secondary 46E30. Mathematics Subject Classification

Positive Toeplitz Operators of Schatten–Herz Type

“…The holomorphic version of the equivalence (a) ⇐⇒ (c) above was proved on the disk in [9]. While our method for the proof of (a) ⇐⇒ (c) is basically adapted from [9], substantial and nontrivial amount of extra work is required for the setting of harmonic Bergman spaces. In the same paper [9], however, the holomorphic version of the equivalence (a) ⇐⇒ (b) was proved only for the case p = 1 or p = q and the case q = p > 1 has been left open.…”

“…While our method for the proof of (a) ⇐⇒ (c) is basically adapted from [9], substantial and nontrivial amount of extra work is required for the setting of harmonic Bergman spaces. In the same paper [9], however, the holomorphic version of the equivalence (a) ⇐⇒ (b) was proved only for the case p = 1 or p = q and the case q = p > 1 has been left open. One can easily modify our argument of the present paper to remove such a restriction as above.…”

Toeplitz Operators and Herz Spaces on the Half-Space

“…See [3], [4], [5] and [8]. Motivated by an earlier work [7] in the holomorphic case on the unit disk, Choe et al [2] have recently studied another aspect of positive Toeplitz operators on the ball. Roughly speaking, they decomposed a given operator into a family of local operators, introduced mixed norm spaces associated with Schatten classes and then gave characterizations of membership in those spaces in terms of the socalled Herz spaces.…”

Boundedness of Berezin Transform on Herz Spaces

“…Previously, in the holomorphic case on the unit disk, Loaiza, López-García and Pérez-Esteva ( [3]) introduced Herz spaces which have mixed norm spaces associated with Schatten classes and they decomposed a given positive Toeplitz operator into a family of local operators and then characterized membership in those spaces. In the harmonic case of unit ball of R n , Choe, Koo and Na ( [2]) showed the boundedness of the Berezin transform on Herz spaces K p,γ q with restricted parameters p and q.…”