Schatten Von Neumann Properties Research Articles

Two-wavelet theory and two-wavelet localization operators on the q-Dunkl harmonic analysis

In this paper, using the [Formula: see text]-Dunkl harmonic analysis introduced in Bettaibi and Bettaieb [q-Analog of the Dunkl transform on the real line, Tamsui Oxf. J. Math. Sci. 25(2) (2007) 117–205] and motivated by Wong’s approach [M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, Berlin, 2002)], we first define and study the [Formula: see text]-wavelets, the continuous [Formula: see text]-wavelet transforms. Next, we introduce the two-wavelet localization operators, the Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented.

Continuity Properties for Born-Jordan Operators with Symbols in Hörmander Classes and Modulation Spaces

We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hormander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus.

Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes

We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in [Formula: see text]. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate Hörmander classes.

Factorizations and Singular Value Estimates of Operators with Gelfand\u2013Shilov and Pilipovi\u0107 kernels

We prove that any linear operator with kernel in a Pilipović or Gelfand–Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten–von Neumann properties for such operators.

Multiple operator integrals, Haagerup and Haagerup‐like tensor products, and operator ideals

We study Schatten–von Neumann properties of multiple operator integrals with integrands in the Haagerup tensor product of L ∞ spaces. We obtain sharp, best possible estimates. This allowed us to obtain sharp Schatten–von Neumann estimates in the case of Haagerup-like tensor products.

Dunkl two-wavelet theory and localization operators

In this paper the notion of a Dunkl two-wavelet is introduced. The resolution of the identity formula for the Dunkl continuous wavelet transform is then formulated and proved. Calderon’s type reproducing formula in the context of the Dunkl two-wavelet theory is proved. The two-wavelet localization operators in the setting of the Dunkl theory are then defined. The Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. It is proved that under suitable conditions on the symbols and two Dunkl wavelets, the boundedness and compactness of these localization operators on \(L^{p}_{k}(\mathbb {R}^{d})\), \(1 \le p \le \infty \). Finally typical examples of localization operators are presented.

Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Grochenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.

On the Schatten–von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten–von Neumann classes Sq with 0<q⩽1. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.

Decompositions of Gelfand–Shilov kernels into kernels of similar class

We prove that any linear operator with a kernel in a Gelfand–Shilov space is a composition of two operators with kernels in the same Gelfand–Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten–von Neumann properties for such operators.

A remark on Schatten–von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂ Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace–Beltrami operator on ∂ Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten–von Neumann class of any order p for which p > dim Ω − 1 3 . Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten–von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Š. Birman on Schatten–von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.

Schatten--von Neumann properties for Fourier integral operators with non-smooth symbols II

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in weighted modulation spaces, when acting on modulation spaces themselves. The phase func ...

Schatten–von Neumann properties for Fourier integral operators with non-smooth symbols, I

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.

Schatten-von Neumann properties of bilinear Hankel forms of higher weights

Hankel forms of higher weights, on weighted Bergman spaces in the unit ball of $\mathsf{C}^d$, were introduced by Peetre. Each Hankel form corresponds to a vector-valued holomorphic function, called the symbol of the form. In this paper we characterize bounded, compact and Schatten-von Neumann $\mathcal{S}_p$ class ($2\leq p&lt;\infty$) Hankel forms in terms of the membership of the symbols in certain Besov spaces.

Characterization of Clifford-valued Hardy spaces and compensated compactness

In this paper, the general Clifford R n,s -valued Hardy spaces and conjugate Hardy spaces are characterized. In particular, each function in R n -valued Hardy space can be determined by half of its function components through Riesz transform, and the explicit determining formulas are given. The products of two functions in the Hardy space give six kinds of compensated quantities, which correspond to six paracommutators, and their boundedness, compactness and Schatten-von Neumann properties are given.

Phase space, wavelet transform and Toeplitz-Hankel type operators

Let IG(n) be the Euclidean group with dilations. It has a maximal compact subgroup SO(n−1). The homogeneous space can be realized as the phase space IG(n)/SO(n−1)≅R n ×R n . The square-integrable representation gives the admissible wavelets AW and wavelet transforms onL 2(R n ). With Laguerre polynomials and surface spherical harmonics an orthogonal decomposition of AW is given; it turns to give a complete orthogonal decomposition of theL 2-space on the phase spaceL 2(R n ×R n ,dxdy/|y| n+1) of the form ⊕ k=0 ∞ ⊕ l=0 ∞ ⊕ j=0 al A l,j k . The Schatten-von Neumann properties of the Toeplitz-Hankel type operators between these decomposition components are established.

Toeplitz and Hankel type operators on Bergman space

The analytic paracommutators in the periodic case have been studied. Their boundedness, compactness, the Schatten-von Neumann properties and the cut-off phenomena have been proved. These results have been applied to some kind of operators on the Bergman spaces that have cut-off at any p∈(0, ∞).

PARACOMMUTATORS ON PRODUCT SPACES

The boundedness and the Schatten-von Neumann properties of paracommutators on product spaces are studied. They turn out to be characterized by the membership of their symbols in the BMO space and the Besov spaces on product spaces.

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measureyαdxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.

Boundedness of paracommutators onL p -spaces

Paracommutator was first introduced and studied by Janson and Peetre in 1985. They mainly discussed its boundedness and Schatten-von Neumann properties onL 2-space. In this paper we study the boundedness of paracommutator onL p -spaces and obtain some useful criteria.