Norm Inequalities Research Articles

Singular Value and Matrix Norm Inequalities between Positive Semidefinite Matrices and Their Blocks

In this paper, we obtain some inequalities involving positive semidefinite 2×2 block matrices and their blocks.

Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type.

On a conjecture related to the geometric mean and norm inequalities

On a conjecture related to the geometric mean and norm inequalities

Extension formulas and norm inequalities in Sobolev Hilbert spaces

Extension formulas and norm inequalities in Sobolev Hilbert spaces

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Interpolation unitarily invariant norms inequalities for matrices with applications

<abstract><p>Let $ A_j, B_j, P_j $, and $ Q_j \in M_{n}(\mathbb{C}) $, where $ j = 1, 2, \dots, m $. For a real number $ c \in [0, 1] $, we prove the following interpolation inequality:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {{A_j}{P_j}{Q_j}^*{B_j}^*} } \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}^2 \leq {\left( {\max \left\{ {L,\,M} \right\}} \right)^4} {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {K_c} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert} \quad {\left\vert\kern-0.1ex\left\vert\kern-0.1ex\left\vert {K_{1-c}} \right\vert\kern-0.1ex\right\vert\kern-0.1ex\right\vert}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} L = {\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {\left| {{A_j}^*} \right|^2} } \right\vert\kern-0.1ex\right\vert}^ \frac{1}{2}, M = {\left\vert\kern-0.1ex\left\vert {\sum\limits_{j = 1}^m {\left| {{B_j}^*} \right|^2} } \right\vert\kern-0.1ex\right\vert}^ \frac{1}{2}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{equation*} K_c = \left( {c{{\left| {{P_1}} \right|}^2} + \left( {1 - c} \right){{\left| {{Q_1}} \right|}^2}} \right) \oplus \cdots \oplus \left( {c{{\left| {{P_m}} \right|}^2} + \left( {1 - c} \right){{\left| {{Q_m}} \right|}^2}} \right). \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>Many other related interpolation inequalities are also obtained.</p></abstract>

Some matrix inequalities related to norm and singular values

<abstract><p>In this short note, we presented a new proof of a weak log-majorization inequality for normal matrices and obtained a singular value inequality related to positive semi-definite matrices. What's more, we also gave an example to show that some conditions in an existing norm inequality are necessary.</p></abstract>

Weighted norm inequalities for Schrödinger operators on variable Lebesgue spaces

Weighted norm inequalities for Schrödinger operators on variable Lebesgue spaces

Two-weight norm inequalities for rough fractional integral operators on Morrey spaces

We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces.

Berezin number and Berezin norm inequalities for operator matrices

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if is an operator matrix with for , then and where if i<j and if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

Singular value and norm inequalities involving the numerical radii of matrices

Singular value and norm inequalities involving the numerical radii of matrices

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Weak log-majorization and inequalities of power means

As noncommutative versions of the quasi-arithmetic mean, we consider the Lim-Pálfia's power mean, Rényi right mean, and Rényi power means. We prove that the Lim-Pálfia's power mean of order $t \in [-1,0)$ is weakly log-majorized by the log-Euclidean mean and fulfills the Ando-Hiai inequality. We establish the log-majorization relationship between the Rényi relative entropy and the product of square roots of given variables. Furthermore, we show the norm inequalities among power means and provide the boundedness of Rényi power mean in terms of the quasi-arithmetic mean.

Minimal [formula omitted]-solutions to singular sublinear elliptic problems

We solve the existence problem for the minimal positive solutions u∈Lp(Ω,dx) to the Dirichlet problems for sublinear elliptic equations of the form Lu=σuq+μinΩ,lim infx→yu(x)=0y∈∂∞Ω,where 0<q<1 and Lu≔−div(A(x)∇u) is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient σ and data μ are nonnegative Radon measures on an arbitrary domain Ω⊂Rn with a positive Green function associated with L. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

Some inequalities about Bourin's question in noncommutative L p -spaces

This paper studies a question of Bourin on noncommutative spaces. Some inequalities due to Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin II. Int J Math. 2021;32:2150043] and Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin III. Positivity. 2022;26:23] are extended to the case of operators in noncommutative spaces associated with a semifinite von Neumann algebra. We give a partial answer to a question of Bourin on noncommutative spaces. Some related submajorization inequalities are also given.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

Weighted norm inequalities related to fractional Schrödinger operators

Weighted norm inequalities related to fractional Schrödinger operators

Mean Inequalities for the Numerical Radius

Extending certain scalar and norm inequalities, we present new inequalities for the numerical radius, which generalize and refine some known results. Applications of the obtained inequalities include a new original proof of the matrix arithmetic-geometric mean inequality and certain extensions of some well-established results from the literature for products of matrices.

Two Weighted Norm Inequalities of Potential Type Operator on Herz Spaces

Two Weighted Norm Inequalities of Potential Type Operator on Herz Spaces