Inequalities For Operators Research Articles

Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators

Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators

On Numerical Radius Inequalities for Hilbert Space Operators

This paper aims to find special cases of some inequalities for numerical radii and spectral radii of a bounded linear operator on a Hil-bert space, we focus on numerical radii inequalities for restricted linear operators on complex Hil-bert spaces for the case of one and two operators, and study the numerical range of an operator K on a complex Hil-bert space H, after that we present some inequalities for numerical radii and spectral radii and studied it to find new results. At the end of this paper we find several inequalities for numerical radii by using the spectral norm, this study is necessary to find other bound for zeros of polynomials and this study is necessary to find new bounds for the zeros of polynomials by a playing the new results to the companion matrix.

Enhancing Adversarial Robustness through Stable Adversarial Training

Deep neural network models are vulnerable to attacks from adversarial methods, such as gradient attacks. Evening small perturbations can cause significant differences in their predictions. Adversarial training (AT) aims to improve the model’s adversarial robustness against gradient attacks by generating adversarial samples and optimizing the adversarial training objective function of the model. Existing methods mainly focus on improving robust accuracy, balancing natural and robust accuracy and suppressing robust overfitting. They rarely consider the AT problem from the characteristics of deep neural networks themselves, such as the stability properties under certain conditions. From a mathematical perspective, deep neural networks with stable training processes may have a better ability to suppress overfitting, as their training process is smoother and avoids sudden drops in performance. We provide a proof of the existence of Ulam stability for deep neural networks. Ulam stability not only determines the existence of the solution for an operator inequality, but it also provides an error bound between the exact and approximate solutions. The feature subspace of a deep neural network with Ulam stability can be accurately characterized and constrained by a function with special properties and a controlled error boundary constant. This restricted feature subspace leads to a more stable training process. Based on these properties, we propose an adversarial training framework called Ulam stability adversarial training (US-AT). This framework can incorporate different Ulam stability conditions and benchmark AT models, optimize the construction of the optimal feature subspace, and consistently improve the model’s robustness and training stability. US-AT is simple and easy to use, and it can be easily integrated with existing multi-class AT models, such as GradAlign and TRADES. Experimental results show that US-AT methods can consistently improve the robust accuracy and training stability of benchmark models.

Generalized Tensorial Simpson type Inequalities for Convex functions of Selfadjoint Operators in Hilbert Space

Several generalized Simpson tensorial type inequalities for self adjoint operators have been obtained with variation depending on the conditions imposed on the function f.

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1<p<\infty $$ 1 < p < ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

Power vector inequalities for operator pairs in Hilbert spaces and their applications

Abstract This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .

On the Inequalities of the Block Power Projection Operators

On the Inequalities of the Block Power Projection Operators

Around strongly operator convex functions

We establish the subadditivity of strongly operator convex functions on (0,∞) and (−∞,0). By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on (−∞,0). We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on (0,∞). Moreover, we study strongly operator convex functions on (a,∞) with −∞<a and on the left half-line (−∞,b) with b<∞. We demonstrate that any nonconstant strongly operator convex function on (a,∞) is strictly operator decreasing, and any nonconstant strongly operator convex function on (−∞,b) is strictly operator monotone. Consequently, for a strongly operator convex function g on (a,∞) or (−∞,b), we provide lower bounds for |g(A)−g(B)| whenever A−B>0.

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

Inequalities for operators and operator pairs in Hilbert spaces

Inequalities for operators and operator pairs in Hilbert spaces

Generalized Choi–Davis–Jensen’s Operator Inequalities and Their Applications

The original Choi–Davis–Jensen’s inequality, known for its extensive applications in various scientific and engineering fields, has inspired researchers to pursue its generalizations. In this study, we extend the Choi–Davis–Jensen’s inequality by introducing a nonlinear map instead of a normalized linear map and generalize the concept of operator convex functions to include any continuous function defined within a compact region. Notably, operators can be matrices with structural symmetry, enhancing the scope and applicability of our results. The Stone–Weierstrass theorem and the Kantorovich function play crucial roles in the formulation and proof of these generalized Choi–Davis–Jensen’s inequalities. Furthermore, we demonstrate an application of this generalized inequality in the context of statistical physics.

On p-numerical radius inequalities for commutators of operators

In this work, we give some p-numerical radius inequalities for operators as well as for commutators of operators. As applications, when p = ∞, we improve some earlier numerical radius inequalities for operators. Also, we refine the triangle inequalities for the numerical radius and the operator norm.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces

Abstract In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

Euclidean operator radius inequalities of d-tuple operators and operator matrices

Abstract We study Euclidean operator radius inequalities of d-tuple operators as well as the sum and the product of d-tuple operators. A power inequality for the Euclidean operator radius of d-tuple operators is also studied. Further, we study the Euclidean operator radius inequalities of 2 × 2 operator matrices whose entries are d-tuple operators.

Some norm inequalities for accretive Hilbert space operators

New norm inequalities for accretive operators on Hilbert space are given. Among other inequalities, we prove that if \(A, B \in \mathbb{B(H)}\) and \(B\) is self-adjoint and also \(C_{m,M}(iAB)\) is accretive, then \begin{eqnarray*} \frac{4 \sqrt{Mm}}{M+m} \Vert AB\Vert \leq \omega(AB-BA^*),\end{eqnarray*} where \(M\) and \(m\) are positive real numbers with \(M &gt; m\) and \(C_{m,M}(A) = (A^* - mI)(MI - A)\). Also, we show that if \(C_{m,M}(A)\) is accretive and \((M-m) \leq k \Vert A \Vert\), then \begin{eqnarray*} \omega(AB) \leq ( 2 + k)\omega(A)\omega(B).\end{eqnarray*}

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Inequalities for Maximal Operators Associated with a Family of General Sets

Inequalities for Maximal Operators Associated with a Family of General Sets

Tensorial Simpson 1/8 Type Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Space

Several Simpson 1 8 tensorial type inequalities for selfadjoint operators have been obtained with variation depending on the conditions imposed on the function f||1/8[f(A)⊗1 + 6f(A⊗1 + 1⊗B/2) + 1⊗f(B)] − ∫01f(λ1⊗B + (1−λ)A⊗1)dλ|| ≤ 5||1⊗B − A⊗1||/32 ||f’||I,+∞.

Using Operator Inequalities in Studying the Stability of Difference Schemes for Nonlinear Boundary Value Problems with Nonlinearities of Unbounded Growth

Using Operator Inequalities in Studying the Stability of Difference Schemes for Nonlinear Boundary Value Problems with Nonlinearities of Unbounded Growth