Inequalities For Operators Research Articles

A unified weighted inequality for fourth-order partial differential operators and applications

In this paper, we establish a fundamental inequality for fourth order partial differential operator P=△α∂s+β∂ss+Δ2 (▪) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator P with some suitable boundary conditions in the case of β<0 or α≠0,β=0. As application, we obtain a resolvent estimate for P, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.

Kantorovich inequality for positive operators on quaternionic Hilbert spaces

Kantorovich inequality for positive operators on quaternionic Hilbert spaces

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

Sharpening celebrated convex inequalities with applications to operators and entropies

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented.

On some generalized numerical radius inequalities for Hilbert space operators

On some generalized numerical radius inequalities for Hilbert space operators

Near-Field Matching and Universal Limits on Electromagnetic Energy Transfer

This article introduces the concept of near-field (NF) matching as a continuum-mode generalization of port matching in circuit theory suitable for field-theoretic electromagnetic energy transfer scenarios, with a focus on spatio-frequency processes in coupled systems. The concept is rigorously formulated using the full electromagnetic Green’s function of a generic receiving surface interacting with arbitrary illumination fields where the Riemannian structure and the electromagnetic boundary condition of the problem are encoded into the tensor structure of a Green’s function on a manifold. After a carefully selected combination of proper function spaces for the various field quantities involved, we utilize exact methods to estimate the sizes of various operator quantities using the appropriate function space norms. A field-theoretic measure of power transfer efficiency in generalized NF matching scenarios is introduced, and exact upper bounds on this efficiency are derived using Young’s inequality for integral kernel operators. This theoretical study complements and generalizes the largely empirical and problem-specific literature on wireless energy transfer by providing an exact and rigorous mathematical framework that can guide and inform future optimization and design processes.

Mixed weak type inequalities for the fractional maximal operator

Mixed weak type inequalities for the fractional maximal operator

Proper improvements of well-known perturbation estimations for the parallel sum

Let A:B be the parallel sum of the positive semidefinite matrices A and B. In this paper, we first establish an operator inequality involving A:B and then apply it to obtain some norm upper bounds for the perturbations of A:B. Our results lead to considerable improvements of the well-known inequalities due to Anderson and Duffin (1969) [2]. Examples are also presented to illustrate the sharpness of the obtained norm upper bounds.

On some inequalities for numerical radius of semi-Hilbert space multioperators

Abstract In this paper, we aim to introduce new inequalities of the m-tuples of semi-Hilbert space operators. These inequalities, which are based on some numerical radius inequalities due to Dragomir et al. for a single operator on a Hilbert space. Further, we give a new formula of the A-Euclidean operator semi-norm of m-tuples of operators on a semi-Hilbert space.

On Further Refinements of Numerical Radius Inequalities

This paper introduces several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. More preciously, these inequalities refine the recent inequalities that were proved in literature. It has already been demonstrated that some inequalities can be improved or restored by concatenating some into one inequality. The main idea of this paper is to extend the existing numerical radius inequalities by providing a unified framework. We also present a numerical example to demonstrate the effectiveness of the proposed approach. Roughly, our approach combines the existing inequalities, proved in literature, into a single inequality that can be used to obtain improved or restored results. This unified approach allows us to extend the existing numerical radius inequalities and show their effectiveness through numerical experiments.

Generalized Operator Shannon Entropy and Related Operator Inequalities

Generalized Operator Shannon Entropy and Related Operator Inequalities

Sharp constants in inequalities admitting the Calderón transference principle

AbstractThe aim of this note is twofold. First, we prove an abstract version of the Calderón transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does not increase the constants in the transferred inequalities. Second, we use the last information to study a certain dichotomy arising in problems of finding the best constants in the weak type $(1,1)$ and strong type $(p,p)$ inequalities for one-parameter ergodic maximal operators.

Norm inequalities for the spectral spread of Hermitian operators

AbstractIn this work, we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self‐adjoint) operator acting on a separable infinite‐dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self‐adjoint operators, that are related to Tao's inequalities for anti‐diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the arithmetic–geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).

Corrigendum to “From $$A_1$$ to $$A_\infty $$: New Mixed Inequalities for Certain Maximal Operators”

Corrigendum to “From $$A_1$$ to $$A_\infty $$: New Mixed Inequalities for Certain Maximal Operators”

A characterization for $B$-singular integral operator and its commutators on generalized weighted $B$-Morrey spaces

We study the maximal operator $M_{\gamma}$ and the singular integral operator $A_{\gamma}$, associated with the generalized shift operator. The generalized shift operators are associated with the Laplace-Bessel differential operator. Our analysis is based on two weighted inequalities for the maximal operator, singular integral operators, and their commutators, related to the Laplace-Bessel differential operator in generalized weighted $B$-Morrey spaces.

On some inequalities for uniformly convex mapping with estimations to normal distributions

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).

Sharp [formula omitted]-error estimates for sampling operators

We study approximation properties of linear sampling operators in the spaces Lp for 1≤p<∞. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in Lp and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in Lp, find the exact order of decay of the corresponding Lp-errors for particular classes of functions, and introduce a special K-functional and its realization suitable for studying smoothness properties of sampling operators.

On isometric embeddability of $S_q^m$ into $S_p^n$ as non-commutative quasi-Banach spaces

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$, has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$, where $(q,p)\in (0,\infty )\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$$\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$$\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty )\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

Global functional calculus, lower/upper bounds and evolution equations on manifolds with boundary

Given a smooth manifold M (with or without boundary), in this paper we establish a global functional calculus, without the standard assumption that the operators are classical pseudo-differential operators, and the Gårding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is free of local coordinate systems. Applications of the Gårding inequality to the global solvability for a class of evolution problems are also considered.

Further Generalizations of Some Fractional Integral Inequalities

This paper aims to establish generalized fractional integral inequalities for operators containing Mittag–Leffler functions. By applying (α,h−m)−p-convexity of real valued functions, generalizations of many well-known inequalities are obtained. Hadamard-type inequalities for various classes of functions are given in particular cases.