Concavity of certain maps on positive definite matrices and applications to Hadamard products

  title={Concavity of certain maps on positive definite matrices and applications to Hadamard products},
  author={T. And{\^o}},
  journal={Linear Algebra and its Applications},
  • T. Andô
  • Published 1979
  • Mathematics
  • Linear Algebra and its Applications
Abstract If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Lowner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: (A,B)↦f[Φ 1 (A) −1 ⊗Φ 2 (B)]·(Φ 1 (A)⊗I) is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map (A,B)↦f[Φ 1 (A)⊗Φ 2 (B) −1 ]·(Φ 1 (A)⊗I) is convex. These yield the concavity of the map (A,B)↦A 1−p ⊗B p (0 (A,B)↦A 1+p ⊗B −p (0 (A,B… Expand
Convex maps on Rn and positive definite matrices
We obtain several convexity statements involving positive definite matrices. In particular, if A,B , X ,Y are invertible matrices and A,B are positive, we show that the map (s, t ) 7→ Tr log(X∗As XExpand
Some results on matrix monotone functions
Abstract Let A and B be Hermitian matrices. We say that A⩾B if A−B is nonnegative definite. A function ƒ:(0,∞) → R is said to be matrix monotone (m.m.) if A⩾B⩾0 implies that ƒ(A) ⩾ ƒ(B) . MatrixExpand
Complementary inequalities to Davis-Choi-Jensen's inequality and operator power means
Let f be an operator convex function on (0,∞), and Φ be a unital positive linear maps on B(H). we give a complementary inequality to Davis-ChoiJensen’s inequality as follows f(Φ(A)) ≥ 4R(A,B) (1Expand
Extensions of Lieb’s Concavity Theorem
AbstractThe operator function (A,B)→ Trf(A,B)(K*)K, defined in pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert SchmidtExpand
A general double inequality related to operator means and positive linear maps
Abstract Let A , B ∈ B ( H ) be such that 0 b 1 I ⩽ A ⩽ a 1 I and 0 b 2 I ⩽ B ⩽ a 2 I for some scalars 0 b i a i , i = 1 , 2 and Φ : B ( H ) → B ( K ) be a positive linear map. We show that for anyExpand
Families of completely positive maps associated with monotone metrics
Abstract An operator convex function on ( 0 , ∞ ) which satisfies the symmetry condition k ( x − 1 ) = x k ( x ) can be used to define a type of non-commutative multiplication by a positive definiteExpand
From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture
Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) s , p , q , s ∈ R , where A and B are positive definiteExpand
Some matrix power and Karcher means inequalities involving positive linear maps
In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1, · · ·Expand
This paper concerns polynomials in g noncommutative variables x=(x1,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which areExpand
On some refinement of the Cauchy-Schwarz inequality
Abstract If A and B are positive semidefinite operators on a Hilbert space and if σ is an operator mean in the sense of Kubo and Ando, then the operator inequality ( A # B ) ⊗ ( A # B ) ⩽ 1 2 { ( A σExpand


Series and parallel addition of matrices
Abstract Let A and B be Hermitian semi-definite matrices and let A + denote the Moore-Penrose generalized inverse. Then we define the parallel sum of A and B by the formula A ( A + B ) + B and denoteExpand
A Schwarz inequality for convex operator functions
For any Hilbert space 3C, let 3C be the totality of bounded selfadjoint operators with spectrum contained in an interval I, which need not be finite. If f is a function from 3C to the self-adjointExpand
On the positive semidefinite nature of a certain matrix expression
By “positive definite matrices” or, briefly, definite matrices, we mean in this note self-adjoint matrices all the characteristic values of which are positive. Alternatively, they can be defined asExpand
Convex trace functions and the Wigner-Yanase-Dyson conjecture
Several convex mappings of linear operators on a Hilbert space into the real numbers are derived, an example being A → — Tr exp(L + In A). Some of these have applications to physics, specifically toExpand
Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory
We show that the Wigner-Yanase-Dyson-Lieb concavity is a general property of an interpolation theory which works between pairs of (hilbertian) seminorms. As an application, the theory extends theExpand
Remarks on two theorems of E. Lieb
The concavity of two functions of a positive matrixA, Tr exp(B + logA) and TrArKApK* (whereB=B* andK are fixed matrices), recently proved by Lieb, can also be obtained by using the theory of HerglotzExpand
Monotone Matrix Functions and Analytic Continuation
I. Preliminaries.- II. Pick Functions.- III. Pick Matrices and Loewner Determinants.- IV. Fatou Theorems.- V. The Spectral Theorem.- VI. One-Dimensional Perturbations.- VII. Monotone MatrixExpand
Functional calculus for sesquilinear forms and the purification map
The paper gives a proposition of a functional calculus for positive sesquilinear forms. A definition of any homogeneous function of two positive sesquilinear forms is given. The purification map forExpand
Hadamard products and multivariate statistical analysis
Abstract The Hadamard product of two matrices multiplied together elementwise is a rather neglected concept in matrix theory and has found only brief and scattered application in statisticalExpand
A Class of Monotone Operator Functions Related to Electrical Network Theory
A new class of monotone functions which map positive operators to positive operators is defined and studied. The class is motivated by electrical network theory. Various properties of these functionsExpand