On some eigenvalue inequalities for Schatten–von Neumann operators

Michael Gil' recently obtained some bounds for eigenvalues in [J. Funct. Anal. 267 (2014) 3500–3506] and [Commun. Contemp. Math. 18 (2016) 1550022], which improve some classical results related to this aspect. We revisit these results by providing genuinely different arguments (e.g., using Aluthge transform, majorization). New results are derived along our discussions.     

Upper Bounds for α-Domination Parameters

We provide a new upper bound for the α-domination number in terms of a parameter α, 0 < α ≤ 1, and graph vertex degrees. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. Using a different probabilistic construction, we prove similar upper bounds for the α-rate domination number, which combines the concepts of α-domination and k-tuple domination.     

Operator-Lipschitz functions in Schatten–von Neumann classes

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S ^α , 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant c _α > 0 such that

On Extrapolation Properties of Schatten–von Neumann Classes

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.     

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-Gröchenig 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.     

von Neumann–Schatten Dual Frames and their Perturbations

In this paper, we discuss the dual of a von Neumann–Schatten p-frames in separable Banach spaces and obtain some of their characterizations. Moreover, we present a classical perturbation result to von Neumann–Schatten p-frames.     

Von Neumann–Schatten Frames in Separable Banach Spaces

In this paper we introduce the notion of a von Neumann-Schatten p-frame in separable Banach spaces and obtain some of their characterizations. We show that p-frames and g-frames are a class of von Neumann-Schatten p-frames.     

Upper and Lower Bounds for Probabilities in the Conditional Borel–Cantelli Lemma

We obtain an inequality connected with a conditional version of the generalized Borel–Cantelli lemma.     

Two Upper Bounds for the Erdős–Szekeres Number with Conditions

We use a projective transformation method of Tóth and Valtr to show that a certain number of points in general position in the plane contain the vertex set of a convex $n$ -gon if their convex hull is an $(n\!-\!1)$ -gon.     

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 prove continuity and Schatten–von Neumann properties of such operators when acting on L ².     

Resolvents of operators inverse to Schatten–von Neumann ones

The paper deals with linear operators in a Hilbert space, whose inverse ones belong to the Schatten–von Neumann ideal of compact operators, and whose imaginary Hermitian components are bounded. A sharp norm estimate for the resolvents of the considered operators is derived. That estimate enables us to investigate spectrum perturbations and to establish bounds for the norms of the semigroups and Hirsch operator functions. The operator logarithm and fractional powers are examples of the Hirsch functions.     

Schatten–von Neumann norm inequalities for two-wavelet localization operators associated with β-Stockwell transforms

In this article we define two-wavelet localization operators corresponding to an irreducible and square-integrable representation of a locally compact Hausdorff group on a Hilbert space. The group structure admitting an irreducible and square-integrable representation which is related to β-Stockwell transform, that we shall use in this article β?∈?R have been introduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis, A group representation related to the Stockwell transform, Indiana Univ. Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norm inequalities of these two-wavelet localization operators are established. The traces and the trace class norm inequalities of the trace class two-wavelet localization operators are given.     

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix

the inequality

(p = 2; 3;?…?) is proved, where λ_k(A) (k = 1; 2;?…?) are the eigenvalues of A and N_r(.) is the norm in SN_r. Moreover, let P(z) = z²I + Bz + C (z ∈ ?) with B ∈ SN_2p, C ∈ SN_p. By z_k(P) (k = 1; 2;?…?) the characteristic values of the pencil P are denoted. It is shown that

In the case p = 1, sharper results are established. In addition, it is derived that

     

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${\mathfrak{S}}_q$ with 0<q?1$0<q?1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.     

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

Periodica Mathematica Hungarica - 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...     

Estimates for the Upper Bounds of the First Eigenvalue on Submanifolds

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Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Balancing Neumann–Neumann methods for the cardiac Bidomain model

Balancing Neumann–Neumann preconditioners are constructed, analyzed and numerically studied for the cardiac Bidomain model in three-dimensions. This reaction–diffusion system is discretized by low-order finite elements in space and implicit–explicit methods in time, yielding very ill-conditioned linear systems that must be solved at each time step. The proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the Bidomain Schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain. The iteration is preconditioned by a Balancing Neumann–Neumann method employing local Neumann solves on each subdomain and a coarse Bidomain solve. A novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for Balancing Neumann–Neumann methods. The theoretical estimates obtained show that the proposed method is scalable, quasi-optimal and robust with respect to possible coefficient discontinuities of the Bidomain operator. The results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory.     

Weighted Bounds for Variational Walsh–Fourier Series

For 1<p<?? and a weight w??A _p and a function in L ^p ([0,1],w) we show that variational sums with sufficiently large exponents of its Walsh?CFourier series are bounded in L ^p (w). This strengthens a result of Hunt?CYoung and is a weighted extension of a variation norm Carleson theorem of Oberlin?CSeeger?CTao?CThiele?CWright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lépingle.