Positive Toeplitz Operators on the Bergman Spaces of the Siegel Upper Half-Space

Abstract

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space. 2010 Mathematics Subject Classification. Primary 47B35; Secondary 32A36.

Schatten class positive Toeplitz operators on Bergman spaces of the Siegel upper half-space

“…In the paper, we are going to study the membership of T μ in Schatten class S p in terms of μ and μ r . It is a natural subsequent work of [7]. Our main result is the following.…”

“…The next lemma is a key result of [7], we shall use it without any explanation, since it is so natural and important.…”

“…Proof Suppose {a k } is an r -lattice such that { μ r (a k )} ∈ l p . First note from [7] that T μ is compact on A 2 (U). We want to show that T μ is in S p .…”

“…where D(z, r ) is the Bergman metric ball at z with radius r and |D(z, r )| denotes the Lebesgue measure of D(z, r ). Quite recently in [7], we have characterized the boundedness and compactness of T μ on Bergman spaces in terms of μ and μ r . In the paper, we are going to study the membership of T μ in Schatten class S p in terms of μ and μ r .…”

The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain

“…Applications of the geometry of Siegel upper/disk domains are found in radar processing [ 12 , 13 , 14 , 15 ] especially for dealing with Toepliz matrices [ 16 , 17 ], probability density estimations [ 18 ] and probability metric distances [ 19 , 20 , 21 , 22 ], information fusion [ 23 ], neural networks [ 24 ], theoretical physics [ 25 , 26 , 27 ], and image morphology operators [ 28 ], just to cite a few.…”