Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Scientiarum Mathematicarum

J-selfadjoint matrix means and their indefinite inequalities

Natália Bebiano, Rute Lemos, Graça Soares

Published May 29, 2024• 0 citations
Publisher

Abstract

Abstract Let J be a non trivial involutive Hermitian matrix. Consider $${\mathbb {C}}^n$$ C n equipped with the indefinite inner product induced by J, $$[x,y]=y^*J x$$ [ x , y ] = y ∗ J x for all $$x,y\in {{\mathbb {C}}}^n,$$ x , y ∈ C n , which endows the matrix algebra $${\mathbb {C}}^{n\times n}$$ C n × n with a partial order relation $$\le ^J$$ ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint $$\alpha $$ α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.

DOI
10.1007/s44146-024-00136-8

Related papers

SOME OPERATOR INEQUALITIES ON SCHUR PRODUCT AND MATRIX WEIGHTED GEOMETRIC MEAN

HUARONG WANG

We complement some properties on operator inequality related to Schur product. Then we discuss the inequalities around the matrix weighted geometric mean, and giveΦ(B]tA) ≤ Φ(B)]tΦ(A);for any positive and unital linear map Φ. Further, if mI ≤ A; B ≤ MI, thenΦ(A]tB) ≥ 4mM(M + m)2 Φ(A)]tΦ(B):

•Jan 1, 2026•0 citations

On the matrix Heron means and Rényi divergences

Trung Hoa Dinh, ‎Raluca Dumitru, Jose A‎. ‎Franco

Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.

Linear and Multilinear Algebra•May 13, 2020•6 citations

Induced metric and matrix inequalities on unitary matrices

H. F. Chau, Chi-Kwong Li, Yiu‐Tung Poon +1 more

Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$ unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on ${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].

arXiv:1107.1047•Journal of Physics A Mathematical and Theoretical•Feb 16, 2012•2 citations

From Majorization to Horn and Unitary-Orbit Inequalities

Juntai Zhou

Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch--Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.

math.FACross-listedarXiv:2609.05854•Sep 8, 2026

Two sample constructions of all the Hilbert spaces for a given operator $H$ of an observable

Miloslav Znojil

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $\Theta$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda)$, the complete $N-$parametric set of metrics $\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined by closed formula.

•Jun 26, 2008•0 citations

The complete menu of eligible metrics for a family of toy Hamiltonians $H \neq H^\dagger$ with real spectra

Miloslav Znojil

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $Θ$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $λ\in (-1,1)$ in matrix $H^{(N)}(λ)$, the complete $N-$parametric set of metrics $Θ^{(N)}_{α_1,α_2,...,α_N}(λ)$ is recurrently defined by closed formula.

arXiv:0806.4295•Jun 26, 2008•0 citations
Abstract Let J be a non trivial involutive Hermitian matrix. Consider Cn{\mathbb {C}}^nCn C n equipped with the indefinite inner product induced by J, [x,y]=y∗Jx[x,y]=y^*J x[x,y]=y∗Jx [ x , y ] = y ∗ J x for all x,y∈Cn,x,y\in {{\mathbb {C}}}^n,x,y∈Cn, x , y ∈ C n , which endows the matrix algebra Cn×n{\mathbb {C}}^{n\times n}Cn×n C n × n with a partial order relation ≤J\le ^J≤J ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint α\alpha α α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.
Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all n×nn\times nn×n unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on Rn{\mathbb R}^nRn induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].
Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch--Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.
Two sample constructions of all the Hilbert spaces for a given operator HHH of an observable
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators Θ\ThetaΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)\lambda \in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(\lambda)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
The complete menu of eligible metrics for a family of toy Hamiltonians H≠H†H \neq H^\daggerH=H† with real spectra
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators ΘΘΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)λ\in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(λ)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)Θ^{(N)}_{α_1,α_2,...,α_N}(λ)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.