Schatten-von Neumann Class Research Articles

Miscellaneous on commutants mod normed ideals and quasicentral modulus $I$

We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K -theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the Mayer–Vietoris sequence. We also make a few remarks about bicommutants mod normed ideals and about the quasicentral modulus for the quasinormed p -Schatten–von Neumann classes 0 < p < 1.

On a class of anharmonic oscillators II. General case

In this work we study a class of anharmonic oscillators on Rn corresponding to Hamiltonians of the form A(D)+V(x), where A(ξ) and V(x) are C∞ functions enjoying some regularity conditions. Our class includes fractional relativistic Schrödinger operators and anharmonic oscillators with fractional potentials. By associating a Hörmander metric we obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the operators A(D)+V(x). This extends the analysis in the first part [1], where the case of polynomial A and V has been analysed.

Natural Lacunae Method and Schatten–Von Neumann Classes of the Convergence Exponent

Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type that contrasts the results by Lidskii, where a sequence of contours of the exponential type was used.

Localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol ς and two windows functions and . We first present some basic properties of the windowed Opdam–Cherednik transform. Then, we use modulation spaces associated with the Opdam–Cherednik transform as appropriate classes for symbols and windows and study the boundedness and compactness of the localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten–von Neumann class.

Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\) pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.

Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

On a class of anharmonic oscillators

In this work we study a class of anharmonic oscillators within the framework of the Weyl-Hörmander calculus. A prototype is an operator on Rn of the form (−Δ)ℓ+|x|2k for k,ℓ integers ≥1. We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the anharmonic oscillator (−Δ)ℓ+|x|2k. In particular we give a simple proof for the main term of the spectral asymptotics of these operators. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups.

The Schatten–von Neumann class associated with the Gabor–Riemann–Liouville operator

In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in the Schatten–von Neumann class. We also give a trace formula when the symbol function is nonnegative.

English

We prove that for normal operators N1, $${N_2} \in {\cal L}({\cal H})$$ , the generalized commutator [N1, N2; X] approaches zero when [N1, N2; [N1, N2; X]] tends to zero in the norm of the Schatten-von Neumann class $${{\cal C}_p}$$ with p > 1 and X varies in a bounded set of such a class.

Localization operators associated with the q-Bessel wavelet transform and applications

This paper deals with the study of the localization operators associated with the [Formula: see text]-Bessel wavelet transform. First, we develop the notion of two [Formula: see text]-Bessel wavelets. Second, motivated by Wong’s approach, we present in our setting, the general theory on the Schatten–von Neumann class, including the boundedness and compactness and Schatten class properties. Next, we prove under suitable conditions on the symbols and two [Formula: see text]-Bessel wavelets, the boundedness and compactness of these localization operators on [Formula: see text], [Formula: see text]. We culminate our study by formulating typical examples of localization operators.

On One Method of Studying Spectral Properties of Non-selfadjoint Operators

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.

Local Price uncertainty principle and time-frequency localization operators for the Hankel–Stockwell transform

For the Hankel–Stockwell transform, the Price uncertainty principle is proved, we define the Localization operators and we study their boundedness and compactness. We also show that these operators belong to the so-called Schatten–von Neumann class.

Localization operators and scalogram associated with the generalized continuous wavelet transform on $\mathbb{R}^d$ for the Heckman–Opdam theory

We consider the generalized wavelet transform Φʷh on R for the Heckman–Opdam theory. We study the localization operators associated with Φʷ h ; in particular, we prove that they are in the Schatten–von Neumann class. Next we introduce some results on the scalogram for this transform

Schur multipliers of Schatten–von Neumann classes Sp

We study in this paper properties of Schur multipliers of Schatten von Neumann classes Sp. We prove that for p≤1, Schur multipliers of Sp are necessarily completely bounded.We also introduce for p≤1 a scale Wp of tensor products of ℓ∞ and prove that matrices in Wp are Schur multipliers of Sp. We compare this sufficient condition with the sufficient condition of membership in the p-tensor product of ℓ∞ spaces.

On the Compactly Solvable Differential Operators for First Order

In this paper, by using the methods of the operator theory, the general form of all compactly solvable extension of the minimal operator which is generated by the first order linear differential operator expression in the Hilbert spaces of vector-functions on finite interval has been found. Later on, the spectrum set of the compactly solvable extensions has been investigated. In addition, the asymptotical behavior of the eigenvalues of any inverse of compactly solvable extension has been studied. Finally, the necessary and sufficient condition for the inverse of the compactly solvable extensions to be belong to Schatten–von Neumann classes has been given.

$$L^p$$–$$L^q$$ time-scale localization operator for the continuous Hankel wavelet transform

For the continuous Hankel wavelet transform, we define the two-wavelet localization operators and we study their $$L^p$$ – $$L^q$$ boundedness and compactness. We also show that these operators are in the Schatten-von Neumann class.

Functions of noncommuting operators under perturbation of class Sp

AbstractIn this article we prove that for , there exist pairs of self‐adjoint operators and and a function f on the real line in the homogeneous Besov class such that the differences and belong to the Schatten–von Neumann class Sp but . A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self‐adjoint operators for any .

Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator

We consider the continuous wavelet transform [Formula: see text] associated with the spherical mean operator. We investigate the localization operators for [Formula: see text], in particular, we prove that they are in the Schatten-von Neumann class. Next, we analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given. Finally, we prove many versions of quantitative uncertainty principles for [Formula: see text].

Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices

In this paper, the boundedness and compactness properties of infinite tridiagonal block operator matrices in the direct sum of Hilbert spaces are studied. The necessary and sufficient conditions for these operators belong to Schatten-von Neumann class are given. Then, the results are supported by applications.

Asymptotics of Eigenvalues for Differential Operators of Fractional Order

Abstract In this paper we deal with a linear combination of a second order uniformly elliptic operator and the Kipriyanov fractional differential operator. We use a novel method based on properties of a real component to study such type of operators. We conduct the classification of the operators by belonging of their resolvent to the Schatten-von Neumann class and formulate the sufficient condition for the completeness of the root functions system. Finally we obtain an asymptotic formula.