Inequalities For Unitarily Invariant Norms Research Articles

Norm inequalities for functions of matrices

In this paper, we prove several spectral norm and unitarily invariant norm inequalities for matrices in which the special cases of our results present some known inequalities. Also, some of our results give interpolating inequalities which are related to well-known recent inequalities.

Singular value and unitarily invariant norm inequalities for matrices

Singular value and unitarily invariant norm inequalities for matrices

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).

Generalization of some unitarily invariant norm inequalities for matrices

Generalization of some unitarily invariant norm inequalities for matrices

General tail bounds for random tensors summation: Majorization approach

In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this research extends previous work by considering the tensors summation tail behavior of the top k-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in Chang (2020). Majorization and antisymmetric tensor product tools are main techniques utilized to establish inequalities for unitarily invariant norms of multivariate tensors. The Laplace transform method is integrated with these inequalities for unitarily invariant norms of multivariate tensors to give us tail bounds estimation for the Ky Fan k-norm for a function of the tensors summation. By restricting different random tensor conditions, we obtain generalized tensor Chernoff and Bernstein inequalities.

Some Singular Value Inequalities for Sector Matrices Involving Operator Concave Functions

In this paper, we give some singular value inequalities for sector matrices involving operator concave function, which are generalizations of some existing results. Moreover, we present some unitarily invariant norm inequalities for sector matrices.

Further unitarily invariant norm inequalities for positive semidefinite matrices

Further unitarily invariant norm inequalities for positive semidefinite matrices

Unitarily invariant norm inequalities for positive semidefinite matrices

In this paper, we prove several unitarily invariant norm inequalities for positive semidefinite matrices. Some of these results give generalizations of earlier known inequalities. Among other applications of our inequalities, we obtain the commutator inequality‖XY−YX‖≤‖X‖‖Y‖+12‖X⁎Y−YX⁎‖for all n × n complex matrices X,Y. Here, ‖⋅‖ denotes the spectral norm.

Some refined Young type inequalities using different weights

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

Singular value and unitarily invariant norm inequalities for sums and products of operators

Singular value and unitarily invariant norm inequalities for sums and products of operators

Unitarily invariant norm inequalities for matrix means

The main target of this article is to present several unitarily invariant norm inequalities which are refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities by convexity of some special functions.

A generalized H\xf6lder-type inequalities for measurable operators

We prove a generalized Hölder-type inequality for measurable operators associated with a semi-finite von Neumann algebra which is a generalization of the result shown by Bekjan (Positivity 21:113–126, 2017). This also provides a generalization of the unitarily invariant norm inequalities for matrix due to Bhatia–Kittaneh, Horn–Mathisa, Horn–Zhan and Zou under a cohyponormal condition.

New proofs on two recent inequalities for unitarily invariant norms

In this short note, we provide alternative proofs for several recent results due to Audenaert (Oper. Matrices 9:475–479, 2015) and Zou (J. Math. Inequal. 10:1119–1122, 2016; Linear Algebra Appl. 552:154–162, 2019).

Norm inequalities involving a special class of functions for sector matrices

In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if is a 2ntimes 2n matrix such that numerical range of Z is contained in a sector region S_{alpha } for some alpha in [0,frac{pi }{2} ) , then, for a submultiplicative function h of the class mathcal{C} and every unitarily invariant norm, we have∥h(|Zij|2)∥≤∥hr(sec(α)|Z11|)∥1r∥hs(sec(α)|Z22|)∥1s,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl\\Vert h \\bigl( \\vert Z_{ij} \\vert ^{2} \\bigr) \\bigr\\Vert &\\leq \\bigl\\Vert h^{r} \\bigl( \\sec (\\alpha ) \\vert Z_{11} \\vert \\bigr) \\bigr\\Vert ^{\\frac{1}{r} } \\bigl\\Vert h^{s} \\bigl( \\sec (\\alpha ) \\vert Z_{22} \\vert \\bigr) \\bigr\\Vert ^{ \\frac{1}{s} }, \\end{aligned}$$ \\end{document} where r and s are positive real numbers with frac{1}{r}+frac{1}{s}=1 and i,j=1,2. We also extend some unitarily invariant norm inequalities for sector matrices.

Improved Young and Heinz operator inequalities for unitarily invariant norms

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn (ℂ): | | | A ν X B 1 − ν + A 1 − ν X B ν 2 | | | ≤ ( 4 r 0 − 1 ) | | | A 1 2 X B 1 2 | | | + 2 ( 1 − 2 r 0 ) | | | ( 1 − α ) A 1 2 X B 1 2 + α ( A X + X B 2 ) | | | , $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 1 4 ≤ ν ≤ 3 4 , α ∈ [ 1 2 , ∞ ) $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r 0 = min{ν, 1 – ν}.

Further interpolation inequalities related to arithmetic-geometric mean, Cauchy-Schwarz and Hölder inequalities for unitarily invariant norms

Further interpolation inequalities related to arithmetic-geometric mean, Cauchy-Schwarz and Hölder inequalities for unitarily invariant norms

Unification of the arithmetic–geometric mean and Hölder inequalities for unitarily invariant norms

In this paper, we obtain an inequality for unitarily invariant norms which unifies the arithmetic–geometric mean inequality and the Hölder inequality for unitarily invariant norms.

More Inequalities for Sector Matrices

Several inequalities are presented for sector matrices. First, an analogue of the AM-GM inequality is established. As applications of this inequality, similar inequalities are presented for singular values and norms. Finally, some unitarily invariant norm inequalities are obtained for sector matrices.

Repetition invariant geometric means

We introduce a class of repetition invariant geometric means and obtain corresponding contractive barycentric maps of integrable Borel probability measures on the Cartan–Hadamard Riemannian manifold of positive definite matrices. They retain most of the properties of the Cartan barycenter and lead to the conclusion that there are infinitely many distinct contractive barycentric maps. Inequalities from the derived geometric means including the Yamazaki inequality and unitarily invariant norm inequalities are presented.

Heinz-type numerical radii inequalities

ABSTRACTThe main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.