Operator Monotone Functions Research Articles

Inertia of non-integer Hadamard powers of a non-negative matrix

ABSTRACTWe prove that if is a conditionally negative definite matrix with all entries positive and is a non-constant operator monotone function, then is also conditionally negative definite. Moreover, is invertible if A is invertible. Next, we prove that if is a symmetric matrix with all entries positive and only one positive eigenvalue, and for all , then also has only one positive eigenvalue and it is invertible if A is invertible. Analogous results are also considered when the diagonal entries of A are all zero and off-diagonal entries are all positive, which extends a result of Reams.

Operator Ky Fan type inequalities

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions. Among other things, we prove that if f:[0,∞)→[0,∞) is an operator monotone function with f(1)=1, f′(1)=μ, and associated mean σ, then for all operators A and B on a complex Hilbert space H such that 0<A,B≤12I, we haveA′∇μB′−A′σB′≤A∇μB−AσB, where I is the identity operator on H, A′:=I−A, B′:=I−B, and ∇μ is the μ-weighted arithmetic mean.

Some results on strongly operator convex functions and operator monotone functions

This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We show also that if t0 is a point in the domain of a continuous function f, then f is operator monotone if and only if (f(t)−f(t0)/(t−t0) is strongly operator convex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of φ∘f when f is operator convex or strongly operator convex.

New sharp inequalities for operator means

ABSTRACTNew sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.

New characterizations of operator monotone functions

If σ is a symmetric mean and f is an operator monotone function on [0,∞), thenf(2(A−1+B−1)−1)≤f(AσB)≤f((A+B)/2). Conversely, Ando and Hiai showed that if f is a function that satisfies either one of these inequalities for all positive operators A and B and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone.In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

Tomographic reconstruction of quantum metrics

In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme is chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to the Petz classification theorem. We show that different metrics can be related through a change in the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such a bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map that connects a monotone metric with a new one, which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.

A unified approach to operator monotone functions

The notion of operator monotonicity dates back to a work by Löwner in 1934. A map F:Sn→Sm is called operator monotone, if A⪰B implies F(A)⪰F(B). (Here, Sn is the space of symmetric matrices with the semidefinite partial order ⪰.) Often, the function F is defined in terms of an underlying simpler function f. Of main interest is to find the properties of f that characterize operator monotonicity of F. In that case, it is said that f is also operator monotone. Classical examples are the Löwner operators and the spectral (scalar-valued isotropic) functions. Operator monotonicity for these two classes of functions is characterized in seemingly very different ways.This work extends the notion of operator monotonicity to symmetric functions f on k arguments. The latter is used to define (generated) k-isotropic mapsF:Sn→S(nk) for any n≥k. Necessary and sufficient conditions are given for f to generate an operator monotone k-isotropic map F. When k=1, the k-isotropic map becomes a Löwner operator and when k=n it becomes a spectral function. This allows us to reconcile and explain the differences between the conditions for monotonicity for the Löwner operators and the spectral functions.

Operator means and positivity of block operators

The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators. It is shown that for positive operators A, B, C and for every non-negative operator monotone function f on $$ (0,\infty )$$ , the block matrix $$\begin{aligned} \left( \begin{array}{cc} f(A) &{} f(C) \\ f(C) &{} f(B) \\ \end{array} \right) \end{aligned}$$ is positive if and only if $$C \le A!B$$ . In particular, if $$C \le A!B$$ then $$\begin{aligned} \left( \begin{array}{cc} A &{} C \\ C &{} B \\ \end{array} \right) , \end{aligned}$$ is positive.

Order Isomorphisms of Operator Intervals

We develop a general theory of order isomorphisms of operator intervals. In this way we unify and extend several known results, among others the famous Ludwig’s description of ortho-order automorphisms of effect algebras and Molnar’s characterization of bijective order preserving maps on bounded observables. Besides proving several new results, one of the main contributions of the paper is to provide self-contained proofs of several known theorems whose original proofs depend on various deep results from functional analysis, operator algebras, and geometry. At the end we will show the optimality of the obtained theorems using Lowner’s theory of operator monotone functions.

Invariance of separability probability over reduced states in 4 × 4 bipartite systems

The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit–qubit and rebit–rebit quantum states with respect to the Hilbert–Schmidt measure. A lot of numerical simulations confirm the and conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of depends on (on single qubit subsystems), moreover it depends only on the Bloch radii (r) of D and it is constant in r. Using the Peres–Horodecki criterion for separability we give a mathematical proof for the probability and we present an integral formula for the complex case which hopefully will help to prove the probability, too. We prove Milz and Strunz’s conjecture for rebit–rebit and qubit–qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function is also studied in detail. We show that even in this setting Milz and Strunz’s conjecture holds true and we give an integral formula for separability probability according to this measure.

More on operator monotone and operator convex functions of several variables

Let C1,C2,…,Ck be positive matrices in Mn and f be a continuous real-valued function on [0,∞). In addition, consider Φ as a positive linear functional on Mn and defineϕ(t1,t2,t3,…,tk)=Φ(f(t1C1+t2C2+t3C3+…+tkCk)), as a k variables continuous function on [0,∞)×…×[0,∞). In this paper, we show that if f is an operator convex function of order mn, then ϕ is a k variables operator convex function of order (n1,…,nk) such that m=n1n2…nk. Also, if f is an operator monotone function of order nk+1, then ϕ is a k variables operator monotone function of order n. In particular, if f is a non-negative operator decreasing function on [0,∞), then the function t→Φ(f(A+tB)) is an operator decreasing and can be written as a Laplace transform of a positive measure.

Corrections and complements to my paper “On a class of operator monotone functions of several variables”

Corrections and complements to my paper “On a class of operator monotone functions of several variables”

Inertia of some special matrices

There have been several articles in literature that study the spectral behaviour of special classes of matrices such as the matrices based on power function given by the matrices , the matrices for positive values of r and positive real numbers . Bhatia and Jain in 2015 and Dyn, Goodman and Micchelli in 1986 have studied the spectral behaviour of and , respectively, for all real values of r. The power function is operator monotone when and operator convex when . It is natural to study the inertia of all these matrices when the power function is replaced by any operator monotone or operator convex function. In the present work, inertia of the matrices and is discussed for non-negative operator monotone and operator convex function f, which further leads to many known and new results.

Some inequalities for operator (p, h)-convex functions

Let p be a positive number and h a function on satisfying for any . A non-negative continuous function f on is said to be operator (p, h)-convex ifholds for all positive semidefinite matrices A, B of order n with spectra in K, and for any . In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen–Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi–Davis–Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.

The converse of the Loewner–Heinz inequality via perspective

The Loewner–Heinz inequality is not only the most essential one in operator theory, but also a fundamental tool for treating operator inequalities. The aim of this paper is to investigate the converse of the Loewner–Heinz inequality in the view point of perspective and generalized perspective of operator monotone and multiplicative functions. Indeed, we give perspective inequalities equivalent to the Loewner–Heinz inequality.

Corrections and Complements to My Paper "On a Class of Operator Monotone Functions of Several Variables"

Corrections and Complements to My Paper "On a Class of Operator Monotone Functions of Several Variables"

Unexpected relations which characterize operator means

We give some characterizations of self-adjointness and symmetricity of operator monotone functions by using the Barbour transform f ↦ t + f 1 + f f \mapsto \frac {t+f}{1+f} and show that there are many non-symmetric operator means between the harmonic mean ! ! and the arithmetic mean ∇ \nabla . Indeed, we show that there exists a non-symmetric operator mean between any two symmetric operator means.

Estimates of operator convex and operator monotone functions on bounded intervals

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the L\"owner--Heinz inequality.

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, C̆ encov, and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter h. Due to the sum rules for the frequency moments of the dynamic structure factor, new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator S are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.

On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function f: R+ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra.