Unitarily Invariant Norm Research Articles

UNITARILY INVARIANT NORM AND <i>Q</i>-NORM ESTIMATIONS FOR THE MOORE–PENROSE INVERSE OF MULTIPLICATIVE PERTURBATIONS OF MATRICES

Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B = D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dagger-A^\dagger\Vert_U$ and $\Vert B^\dagger-A^\dagger\Vert_Q$ are derived, where $A^\dagger,B^\dagger$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.

Some norm inequalities for operators

For any unitarily invariant norm on Hilbert-space operators, we prove Holder and Cauchy–Schwarz inequalities. As a consequence, several inequalities are lifted to the operator settings. Some more associated, norm inequalities for operators are obtained.

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

A new young type inequality involving Heinz mean

In this paper, we gave a new Young type inequality and the relevant Heinz mean inequality. Furthermore, we also improved some inequalities with Kantorovich constant or Specht?s ratio. Meanwhile, on the base of our scalars results, we obtain some new corresponding operator inequalities and matrix versions including Hilbert-Schmidt norm, unitarily invariant norm and related trace versions, which can be regarded as the application of our scalar results.

Lieb functions and sectorial matrices

In this article, we present a special treatment of sectorial matrices under Lieb functions. This treatment results from a new characterization of sectorial matrices using blocks, which then implies a new set of inequalities governing such matrices, where singular values, determinants, numerical radius and unitarily invariant norms inequalities are retrieved in a general form that implies many well known results in the literature.

Generalized frame operator distance problems

Let S∈Md(C)+ be a positive semidefinite d×d complex matrix and let a=(ai)i∈Ik∈R>0k, indexed by Ik={1,…,k}, be a k-tuple of positive numbers. Let Td(a) denote the set of families G={gi}i∈Ik∈(Cd)k such that ‖gi‖2=ai, for i∈Ik; thus, Td(a) is the product of spheres in Cd endowed with the product metric. For a strictly convex unitarily invariant norm N in Md(C), we consider the generalized frame operator distance function Θ(N,S,a) defined on Td(a), given byΘ(N,S,a)(G)=N(S−SG)whereSG=∑i∈Ikgigi⁎∈Md(C)+. In this paper we determine the geometrical and spectral structure of local minimizers G0∈Td(a) of Θ(N,S,a). In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of N.

An extension of inequalities by Ando

We give variations on Ando's result comparing f(B)−f(A) and f(|B−A|) with respect to unitarily invariant norms on matrices.

A Beurling theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras with unitarily invariant norms

We prove a Beurling-type theorem for H∞-invariant spaces of Lα(M,τ), where α is a unitarily invariant, locally ∥⋅∥1-dominating, mutually continuous norm with respect to τ, where M is a von Neumann algebra with a faithful, normal, semifinite tracial weight τ, and H∞ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the H∞-invariant subspaces of a noncommutative Banach function space I(τ) with the norm ∥⋅∥E on M, the crossed product of a semifinite von Neumann algebra by an action β, and B(H) for a separable Hilbert space H.

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60−89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.

Interpolating the Rotfel'd inequality for unitarily invariant norms

Let A and B be square matrices of the same order. Uchiyama proved that for all unitarily invariant norms ‖⋅‖,‖f(|A+B|)‖≤‖f(|A|)‖+‖f(|B|)‖, which extended the Rotfel'd Trace Inequality Tr(f(|A+B|))≤Tr(f(|A|))+Tr(f(|B|)) for any concave function f:[0,∞)→[0,∞). In this note, we interpolate Uchiyama's inequality by proving that for all unitarily invariant norms ‖⋅‖,‖f(|A+B|)‖≤‖f(12(|A|+|B|A⁎+B⁎A+B|A⁎|+|B⁎|))‖≤‖f(|A|)‖+‖f(|B|)‖. For instance letting f(t)=t yields‖A+B‖≤‖12(|A|+|B|A⁎+B⁎A+B|A⁎|+|B⁎|)‖≤‖A‖+‖B‖, which interpolates the triangle inequality for norms.

More inequalities for positive semidefinite 2 × 2 block matrices and their blocks

It is obtained that for positive semidefinite block matrix H=[MKK⁎N] if A,B∈Mn(C) such that max(‖A‖2,‖B‖2)≤2 and f is a nonnegative increasing convex function on [0,∞) satisfying f(0)=0, then2sj[f(|A⁎KB|)]≤max(‖A‖2,‖B‖2)sj[f(H)] for j=1,2,…,n. Also, it is shown that if H=[MKK⁎N] is a positive semidefinite 2×2 block matrix with square matrices M and N of the same size, thenHr≤2(a+b)r−1(M⊕N) for r≥1, where a and b are the largest eigenvalues of the matrices M and N, respectively. In addition, if r≥32, then|||Hr|||2≤4|||M2⊕N2|||(|||M⊕0|||+|||N⊕0|||)2r−2 for every unitarily invariant norm.

A note on sectorial matrices

ABSTRACT Assume that the numerical range of is a subset of the sector , for some . It is proved that for some unitary . As a consequence, we prove the following singular value inequalities where is the greatest integer . In the case where partitioned as the following log-majorization inequality is proved for . As a result, we get the following Hölder type inequality for any unitarily invariant norm . Here, r,p and q are positive numbers such that . Related inequalities are also proved.

Randomized Subspace Iteration: Analysis of Canonical Angles and Unitarily Invariant Norms

This paper analyzes the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the exact and the approximate singular subspaces. Second, we derive bounds for the low-rank approximation in any unitarily invariant norm (including the Schatten-p norm). This generalizes the bounds for spectral and Frobenius norms found in the literature. Third, we present bounds for the accuracy of the singular values. The bounds are structural in that they are applicable to any starting guess, be it random or deterministic, that satisfies some minimal assumptions. Specialized bounds are provided when a Gaussian random matrix is used as the starting guess. Numerical experiments demonstrate the effectiveness of the proposed bounds.

Maximum Principles for Normal Matrices

Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.

Non-Linear Interpolation of the Harmonic–Geometric–Arithmetic Matrix Means

Bhatia, Lim, and Yamazaki conjectured that the Kubo–Ando extensions of means of numbers satisfy a norm minimality condition with respect to unitarily invariant norms. In this short note, we introduce a symmetric Kubo–Ando mean and a non-Kubo–Ando extension that do not satisfy this property.

On the convexity of functions

Let A,B, and X be bounded linear operators on a separable Hilbert space such that A,B are positive, X ? ?I, for some positive real number ?, and ? ? [0,1]. Among other results, it is shown that if f(t) is an increasing function on [0,?) with f(0) = 0 such that f(?t) is convex, then ?|||f(?A + (1-?)B) + f(?|A-B|)|||?|||?f(A)X + (1-?)Xf (B)||| for every unitarily invariant norm, where ? = min (?,1-?). Applications of our results are given.

Two inequalities of unitarily invariant norms for matrices

Two inequalities of unitarily invariant norms for matrices

Singular value inequalities for sector matrices

In this paper, we present two singular value inequalities for sector matrices. As a consequence, we prove unitarily invariant norm inequalities for sector matrices. Moreover, we present some determinant inequalities for accretive-dissipative matrices.

Norm inequalities related to the Heinz means

Let (I,|!|!|cdot|!|!|) be a two-sided ideal of operators equipped with a unitarily invariant norm |!|!| cdot|!|!|. We generalize the results of Kapil’s, using a new contractive map in I to obtain a norm inequality. And we give a new inequality, which is a comparison between the Heinz means and other related inequalities; moreover, we will obtain some correlative conclusions.

A note on the $C$ -numerical radius and the $\lambda$ -Aluthge transform in finite factors

We prove that for any two elements A, B in a factor M, if B commutes with all the unitary conjugates of A, then either A or B is in CI. Then we obtain an equivalent condition for the situation that the C-numerical radius ωC(⋅) is a weakly unitarily invariant norm on finite factors, and we also prove some inequalities on the C-numerical radius on finite factors. As an application, we show that for an invertible operator T in a finite factor M, f(△λ(T)) is in the weak operator closure of the set {∑i=1nziUif(T)Ui∗∣n∈N,(Ui)1≤i≤n∈U(M),∑i=1n|zi|≤1}, where f is a polynomial, △λ(T) is the λ-Aluthge transform of T, and 0≤λ≤1.