# A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

Authors: Pattrawut Chansangiam
Journal: International Journal of Analysis
 (2015)

---

## Abstract

This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.


---

## Body

## 1. Introduction

A useful and important class of real-valued functions is the class of operator monotone functions. Such functions were introduced by Löwner in a seminal paper [1]. These functions are functions of Hermitian matrices/operators preserving order. In that paper, he established a relationship between operator monotonicity, the positivity of matrix of divided differences, and an important class of analytic functions, namely, Pick functions. This concept is closely related to operator convex/concave functions which was studied afterwards by Kraus in [2]. Operator monotone functions and operator convex/concave functions arise naturally in matrix and operator inequalities (e.g., [3–7]). This is because the theory of inequalities depends heavily on the concepts of monotonicity, convexity, and concavity. One of the most beautiful and important results in operator theory is the so-called Löwner-Heinz inequality (see [1, 8]) which is equivalent to the operator monotonicity of the function t↦tp for t⩾0 when p∈[0,1]. See more information about operator monotonicity/convexity in [5, Chapter V], [9, Section  2], [10, Chapter  4], and [11].Operator monotone functions have applications in many areas, including functional analysis, mathematical physics, information theory, and electrical engineering; see, for example, [12–15]. This concept plays major roles in the so-called Kubo-Ando theory of operator connections and operator means. This axiomatic theory was introduced in [16] and plays central role in operator inequalities, operator equations, network theory, and quantum information theory. Indeed, there is a one-to-one correspondence between operator monotone functions on the nonnegative reals R+ and operator connections. See more information about applications of operator monotone functions to theory of operator means in [9, 10, 17–19].In this paper, we survey significant results of operator monotone functions related to operator convexity and operator means. We give various characterizations in several viewpoints. The first viewpoint is differential analysis in terms of matrix of divided differences. In viewpoint of operator inequalities, Hansen and Pedersen provide characterizations and relationship of operator monotonicity and operator convexity/concavity. From measure theory viewpoint, every operator monotone function on the nonnegative reals always occurs as an integral of suitable operator monotone functions with respect to a Borel measure. Such functions form building blocks for arbitrary operator monotone functions onR+. A deep theory of Kubo and Ando states that each operator monotone function f on R+ corresponds to a unique operator connection. Moreover, if f(1)=1, then f is associated with an operator mean.Here is the outline of the paper. In Section2, after setting basic notations, we give the definitions and examples of operator monotone/convex functions and provide their characterizations with respect to matrix of divided differences. Section 3 deals with Hansen-Pedersen characterizations of operator monotone/convex functions. The aims of Section 4 are to characterize operator monotone functions on the nonnegative reals in terms of Borel measures and to give some concrete examples. Finally, we establish the correspondence between operator monotone functions and operator means in Section 5.


## 2. Operator Monotonicity and Convexity

Throughout this paper, letH be a complex Hilbert space. Denote by B(H) the algebra of bounded linear operators on H. The spectrum of an operator A is denoted by σ(A). Consider the real vector space B(H)sa of self-adjoint operators on H and its positive cone B(H)+ of positive operators on H. A partial order is naturally equipped on B(H)sa by defining A⩽B if and only if B-A∈B(H)+. We write A>0 to mean that A is a strictly positive operator, or equivalently, A⩾0 and A is invertible. When H=Cn, we identify B(H) with the algebra Mn of n-by-n complex matrices. Then Mn+ is just the cone of n-by-n positive semidefinite matrices.LetA∈Mn be normal. By the spectral resolution of A, there exist distinct scalars λ1,…,λm∈C and projections P1,…,Pm on Cn such that(1)A=∑i=1mλiPi,PiPj=0fori≠j,∑i=1mPi=I.Moreover, these scalars and projections are uniquely determined. In fact, σ(A) is the set {λ1,…,λm} (not counting multiplicities) and each Pi is the projection onto the eigenspace ker(A-λiI). For each function f:σ(A)→C, we can define thefunctional calculus for the function f by(2)fA=∑i=1mfλiPi.Definition 1.
LetJ⊆R be an interval. A function f:J→R is said to be as follows:(i)
Matrix monotone of degreen or n-monotone if, for every A,B∈Mnsa with σ(A),σ(B)⊆J, it holds that(3)A⩽B⟹fA⩽fB.(ii)
Operator monotone ormatrix monotone if it is n-monotone for all n∈N.(iii)
Matrix convex of degreen or n-convex if, for every A,B∈Mnsa with σ(A),σ(B)⊆J, it holds that(4)ftA+1-tB⩽tfA+1-tfB.(iv)
Operator convex if it is n-convex for every n∈N.(v)
Matrix concave of degreen or n-concave if -f is n-convex.(vi)
Operator concave if it is n-concave for every n∈N.Recall that a continuous functionf:J→R isconvex (concave) if and only if it ismidpoint-convex (midpoint-concave, resp.). It follows that a continuous function f:J→R is n-convex if and only if it is n-midpoint convex; that is, the condition (4) holds for t=1/2. In particular, if f is continuous, then f is operator convex if and only if it is operator midpoint-convex. Similar results are applied for the case of concavity.Everyn-monotone function is (n-1)-monotone but the converse is false in general. The condition of being 1-monotone is the monotone increasing in usual sense. The set of operator monotone functions on the interval J is closed under taking nonnegative linear combinations, pointwise limits, and compositions. The straight line t↦mt+c is operator concave and operator convex on the real line for any m,c∈R. This function is operator monotone if and only if the slope m is nonnegative.Proposition 2.
On(0,∞), the function t↦1/t is operator convex and t↦-1/t is operator monotone. On (-∞,0), the function t↦1/t is operator concave and t↦-1/t is operator monotone.Proof.
IfA⩾B>0, then A-1⩽B-1 and hence -A-1⩾-B-1. This shows the operator monotonicity of t↦-1/t on (0,∞). The scalar inequality [(1+t)/2]-1⩽(1+t-1)/2 implies that for every C>0(5)I+C2-1⩽I+C-12.For A,B>0 in Mn, by setting C=A-1/2BA-1/2; we have(6)A+B2-1=A1/2I+CA1/22-1=A-1/2I+C2-1A-1/2⩽A-1/2I+C-12A-1/2=A-1+B-12.Hence t↦1/t is operator convex. For the case (-∞,0), consider -A and -B instead of A and B.It follows from this proposition that the functiont↦(c-t)-1 is operator monotone on (a,b) for any c∉(a,b). The next result is called theLöwner-Heinz inequality. It was first proved by Löwner [1] and also by Heinz [8]. There are many proofs of this fact. The following is due to Pedersen [20].Theorem 3.
ForA⩾B in Mn+ and r∈[0,1], one has Ar⩾Br.Proof.
The continuity argument allows us to considerA⩾B>0. Since p↦Ap and p↦Bp are continuous, the set (7)Ω=p∈R:Ap⩾Bpis closed. Clearly, 0,1∈Ω. Recall that the set of dyadic numbers in [0,1] is dense in [0,1]. Hence, to prove that [0,1]⊆Ω, it suffices to show that(8)p,q∈Ω⟹p+q2∈Ω.Suppose Ap⩾Bp and Aq⩾Bq. Then A-p/2BpA-p/2⩽I and(9)Bp/2A-p/22=Bp/2A-p/2∗Bp/2A-p/2=A-p/2BpA-p/2⩽1.Hence, Bp/2A-p/2⩽1 and similarly Bq/2A-q/2⩽1. Thus (10)1⩾Bp/2A-p/2∗Bq/2A-q/2=A-p/2Bp+q/2A-q/2⩾rA-p/2Bp+q/2A-q/2=rA-p+q/4Bp+q/2A-p+q/4=A-p+q/4Bp+q/2A-p+q/4.Here, r(·) denotes the spectral radius. Now, (11)I⩾A-p+q/4Bp+q/2A-p+q/4or A(p+q)/2⩾B(p+q)/2; that is, (p+q)/2∈Ω.Proposition 4.
For eachp>1, the function t↦tp is not operator monotone on R+.Proof.
ConsiderA=3/42001 and B=1/21111. Then A⩾B⩾0. Since B is a projection, for each p>0, we have Bp=B and(12)Ap-Bp=32p-12-12-1234p-12.Compute (13)det⁡Ap-Bp=38p3p-2p+4p2.If Ap⩾Bp, we must have det⁡(Ap-Bp)⩾0; that is, (2p+4p)/2⩽3p, which is false when p>1.Operator monotone functions can be defined in the context of operators acting on a Hilbert space as illustrated in the next theorem. This is why we also call a matrix monotone function an operator monotone function. Note that in this theorem we assume the continuity off since we need to define the continuous functional calculus of an operator.Theorem 5.
The following statements are equivalent for a continuous functionf:(a,b)→R:(i)
A⩽B⇒f(A)⩽f(B) for all Hermitian matrices A,B of all orders whose spectra are contained in (a,b);(ii)
A⩽B⇒f(A)⩽f(B) for all Hermitian operators A,B∈B(H) whose spectra are contained in (a,b) and for an infinite-dimensional Hilbert space H;(iii)
A⩽B⇒f(A)⩽f(B) for all Hermitian operators A,B∈B(H) whose spectra are contained in (a,b) and for all Hilbert spaces H.Proof.
It is obvious that (iii) implies (ii). The implication (ii)⇒ (i) follows by taking an n-dimensional subspace.
(i)⇒ (iii). For each finite-dimensional subspace F of H, let PF be the orthogonal projection onto F. Suppose that A⩽B in B(H) with spectra in (a,b). Consider nets AF≔PFAPF+c(I-PF) and BF≔PFBPF+c(I-PF) in B(H), where c∈(a,b) is fixed and a directed set:(14)F:Fis a finite-dimensional subspace ofHwith respect to the set inclusion. Since AF→A and BF→B in the strong-operator topology, we have f(AF)→f(A) and f(BF)→f(B) in the strong-operator topology. Note that (15)fAF=fPFAPF+fcI-PF,where f(PFAPF) is the functional calculus of PFAPF in B(F). Since B(F) is identified with Mn with n=dimF and since PFAPF⩽PFBPF as elements of B(F), the assumption (i) implies that f(PFAPF)⩽f(PFBPF) and hence f(AF)⩽f(BF). By taking the limit in the strong-operator topology, we have f(A)⩽f(B).


## 3. Differential Analysis of Operator Monotonicity and Convexity

In this section, we consider topological properties of operator monotone/convex functions. Note that we do not impose a topological assumption on anyn-monotone/concave function. The following three theorems show that any n-monotone/concave function on an open interval is at least continuously differentiable (i.e., C1) function when n⩾2.Theorem 6.
Iff is a 2-monotone function on (a,b), then f is C1 on (a,b) and f′>0 unless f is a constant. In particular, every operator monotone function on (a,b) is C1.Proof.
The proof is very long and it consists of many details. The original proof is contained in [1]; see also [9, Section  2].Theorem 7.
Letn⩾2 be an integer. The following statements are equivalent for a function f:(a,b)→R:(1)
f is n-monotone on (a,b);(2)
f is C1 on (a,b) and [f[1](λi,λj)]i,j=1n⩾0 for every choice of λ1<λ2<⋯<λn from (a,b).Here, the 1st divided difference f[1] is defined as follows:(16)f1x,y≔fx-fyx-y,x≠y,and f1(x,x)≔f′(x).Proof.
See [1] or [9, Section  2].Theorem 8.
Letn⩾2 be an integer. The following statements are equivalent for a function f:(a,b)→R:(1)
f is n-convex on (a,b);(2)
f is C2 on (a,b) and [f[2](λ1,λi,λj)]i,j=1n⩾0 for every choice of λ1,λ2,…,λn from (a,b).Moreover, if f is operator convex, then f[1](λ,·) is operator monotone for every λ∈(a,b). Here, the 2nd divided difference f[2] is defined to be(17)f2x,y,z≔f1x,y-f1y,zx-z,x≠z,and f2(x,y,x)≔f′′(x).Proof.
See [2] or [9, Section  2].


## 4. Hansen-Pedersen Characterizations

In this section, we characterize operator monotone functions in the sense of Hansen-Pedersen [21].Theorem 9.
Letf:[0,α)→R be a function where 0<α⩽∞. Then the following are equivalent:(i)
f is operator convex on [0,α) and f(0)⩽0;(ii)
f is operator convex on (0,α) and f(0+)⩽f(0)⩽0, where the existence of f(0+):=limt→0+f(t) and f(0+)⩽f(0) are automatic from the operator convexity of f on (0,α);(iii)
f(t)/t is operator monotone on (0,α) and f(0+)⩽f(0)⩽0, where the existence of f(0+) and f(0+)⩽f(0) are automatic from the operator monotonicity of f(t)/t on (0,α);(iv)
f(X∗AX)⩽X∗f(A)X for every A∈Mnsa with σ(A)⊂[0,α), for every X∈Mn with X⩽1 and for every n∈N;(v)
f(X∗AX+Y∗BY)⩽X∗f(A)X+Y∗f(B)Y for every A,B∈Mnsa with σ(A),σ(B)⊆[0,α), for every X,Y∈Mn, with X∗X+Y∗Y⩽I and for every n∈N;(vi)
f(PAP)⩽Pf(A)P for every A∈Mnsa with σ(A)⊆[0,α), for every orthogonal projection P on Cn and for every n∈N.Proof.
See [21] or [9, Theorem  2.5.2].Theorem 10.
Ifα=∞ and f(t)⩽0 for all t∈[0,∞), then the conditions of Theorem 9 are also equivalent to(vii)
-f being operator monotone on [0,∞).Proof.
See [21] or [9, Theorem  2.5.3].Corollary 11.
A function onf:[0,∞)→[0,∞) is operator monotone if and only if f is operator concave.Proof.
This is the equivalence between (i) and (vii) of Theorem10.Corollary 12.
Consider the following statements for a functionf:(0,∞)→(0,∞):(i)
f is operator monotone;(ii)
t/f(t) is operator monotone;(iii)
f is operator concave;(iv)
1/f(t) is operator convex.We have (i) ⇔ (ii) ⇔ (iii) ⇒ (iv).Proof.
(i)⇒ (ii). For any ϵ>0, f(t+ϵ) is operator monotone on [0,∞). Theorem 10 implies that -f(t+ϵ)/t is operator monotone on (0,∞). Proposition 2 then implies that(18)tft+ϵ=--ft+ϵt-1is operator monotone on (0,∞). Letting ϵ↘0 yields (ii).
(ii)⇒ (i). For any ϵ>0, (t+ϵ)/f(t+ϵ) is operator monotone on [0,∞). Theorem 10 implies that -(t+ϵ)/tf(t+ϵ) is operator monotone on (0,∞). Proposition 2 then implies that(19)tft+ϵt+ϵ=--t+ϵtft+ϵ-1is operator monotone on (0,∞). Letting ϵ↘0 yields (i).
(i)⇔ (iii). By Corollary 11, we have that(20)fis operator monotone on0,∞⟺ft+ϵis operator monotone on0,∞for anyϵ>0⟺ft+ϵis operator concave on0,∞for anyϵ>0⟺fis operator concave on0,∞.(iii) ⇔ (iv). Write g(t)=1/f(t). Let A,B>0 in Mn. By (iii),(21)fA+B2⩾fA+fB2.Then Proposition 2 implies(22)gA+B2=fA+B2-1⩽fA+fB2-1⩽fA-1+fB-12=gA+gB2.Hence g is operator convex.Example 13.
Consider the following:(i)
For eachp∈[0,1], tp is operator concave on [0,∞).(ii)
The functionf(t)=(t-1)/log⁡t on [0,∞) where f(0)≔0 and f(1)≔1.(iii)
The logarithmic function is operator monotone and operator concave on(0,∞).(iv)
The functiongt=tlog⁡t is operator convex on [0,∞).Proof.
(i) It follows from the Löwner-Heinz inequality and Corollary11. (ii) Use the integral representation f(t)=∫01txdx for t⩾0. (iii) By (ii), t/log⁡(1+t) is operator monotone function on (0,∞). Corollary 12 then implies that log⁡(1+t) is operator monotone and operator concave on (0,∞). Now, for each ϵ>0, log⁡(ϵ+t)=log⁡ϵ+log⁡(1+ϵ-1t) is operator monotone and operator concave on (0,∞). Letting ϵ↘0 yields the result. (iv) Since g is continuous on [0,∞) and g(t)/t=log⁡t is operator monotone on (0,∞), we can conclude that g is operator convex on [0,∞) by Theorem 9.Example 14.
In information theory, the functionft=-tlog⁡t is known as theentropy function. In [22], it was shown that this function is operator concave. An analogue notion of the entropy function in quantum mechanics is the entropy of a density matrix (a positive semidefinite matrix with trace 1) or a positive contraction on a Hilbert space. More precisely, for each positive operator A with A⩽1, we define theentropy of A by (23)SA=-Alog⁡A⩾0.More concrete examples of operator monotone functions are provided in [23].


## 5. Integral Representations of Operator Monotone Functions on the Nonnegative Reals

In this section, we focus on the class of operator monotone functions fromR+ to R+, denoted by OM(R+). A reformulation of Löwner’s theorem (see [1] or [5, Chapter V]) states that every f∈OM(R+) admits an integral representation with respect to Borel measure on the unit interval as follows.Theorem 15 (see [24]).
Given a finite Borel measureμ on [0,1], the function(24)fx=∫0,11!txdμt,x⩾0is an operator monotone function from R+ to R+. In fact, the map μ↦f is bijective, affine, and order-preserving.Thus the functionsx↦1!tx for t∈[0,1] form building blocks for arbitrary operator monotone functions on R+. The measure μ in the previous theorem is called theassociated measure of the operator monotone function f. Moreover, a function f∈OM(R+) is normalized (in the sense that f(1)=1) if and only if μ is a probability measure [24]. This means that every normalized operator monotone function on R+ can be viewed as an average of the special operator monotone functions x↦1!tx for t∈[0,1]. The functions x↦1!tx for t∈[0,1] are extreme points of the convex set of normalized operator monotone functions from R+ to R+.Example 16 (typical examples of “singularly discrete” operator monotone functions).
The operator monotone functionx↦1!tx corresponds to the Dirac measure δt at t∈[0,1]. In particular, the operator monotone functions x↦1 and x↦x correspond to the measures δ0 and δ1, respectively. By affinity of the map μ↦f, the measure ∑i=1naiδti, where ti∈[0,1] and ai⩾0, is associated to the function x↦∑i=1nai(1!tix).Example 17.
The following examples illustrate the associated measures of “absolutely continuous” operator monotone functions; see [24] for details of proofs.(1)
For each0<α<1, the associated measure of the operator monotone function xα is given by(25)dμt=sin⁡αππ·1t1-α1-tαdt.(2)
The associated measure off(x)=(x-1)/log⁡x is dμ(t)=g(t)dt where density function g given by(26)gt=1t1-tπ2+log2⁡t/1-t.(3)
Consider the operator monotone function(27)x⟼1f1/x=xx-1log⁡x.This function has Lebesgue measure as the associated measure, equivalently; we have the integral representation(28)xx-1log⁡x=∫0,11!txdt.Iff∈OM(R+) has μ as the associated measure, then thetranspose of f, defined by x↦xf(1/x), also belongs to OM(R+) and has μΘ as its associated measure. Here, Θ:[0,1]→[0,1] is a homeomorphism defined by t↦1-t.We say thatf issymmetric if it coincides with its transpose. A Borel measure μ on [0,1] is said to besymmetric if μ is invariant under Θ; that is, μΘ=μ.Corollary 18 (see [24]).
There is a one-to-one correspondence between symmetric operator monotone function fromR+ to R+ and finite symmetric Borel measures via the integral representation:(29)fx=12∫0,11!tx+x!t1dμt,x⩾0.In particular, an operator monotone function on R+ is symmetric if and only if its associated measure is symmetric.It follows that there is a one-to-one correspondence between normalized symmetric operator monotone functions onR+ and probability symmetric Borel measures on the unit interval via the integral representation (29).The integral representation (24) also has advantages in treating decompositions of operator monotone functions (see [24]). It turns out that every function f∈OM(R+) can be expressed as (30)f=fac+fsd⁡+fsc,where fac, fsd⁡, and fsc also belong to the class OM(R+). The “singularly discrete part” fsd⁡ is a countable sum of x↦1!tx for t∈[0,1] with nonnegative coefficients. The “absolutely continuous part” fac has an integral representation with respect to Lebesgue measure m on [0,1]. The “singularly continuous part” fsc has an integral representation with respect to a continuous measure mutually singular to m.


## 6. Operator Monotone Functions and Operator Means

This section explains the one-to-one correspondence between operator monotone functions onR+ and operator means.An axiomatic of operator means was investigated by Kubo and Ando [16]. Recall that anoperator connection is a binary operation σ on B(H)+ such that for all positive operators A,B,C,D:(M1)
monotonicity:A⩽C,  B⩽D⇒AσB⩽CσD;(M2)
transformer inequality:C(AσB)C⩽(CAC)σ(CBC);(M3)
continuity from above: forAn,Bn∈B(H)+, if An↓A and Bn↓B, then AnσBn↓AσB. Here, Xn↓X indicates that (Xn) is a decreasing sequence converging strongly to X.Anoperator mean is an operator connection σ with property that AσA=A for all A⩾0, or, equivalently, IσI=0.From the monotonicity (M1) and the continuity (M3), every operator connection is uniquely determined on the set of strictly positive operators. Indeed, once an operator connectionσ is defined for any A,B>0, we have (31)AσB=limϵ↓0⁡A+ϵIσB+ϵI;here the limit is taken in the strong-operator topology.A major core of Kubo-Ando theory is the one-to-one correspondence between operator connections and operator monotone functions.Theorem 19 (see [16, Theorem  3.4]).
Given an operator connectionσ, there is a unique operator monotone function f:R+→R+ such that(32)fA=IσA,A⩾0.In fact, the map σ↦f is a bijection. Moreover, σ is a mean if and only if μ is a probability measure.The functionf in this theorem is called therepresenting function of the operator connection σ. It turns out that, for any A,B>0,(33)AσB=A1/2fA-1/2BA-1/2A1/2.From the integral representation of operator monotone functions (24), every operator connection admits the integral representation(34)AσB=∫01A!tBdμt,A,B⩾0.Since the weighted harmonic means are jointly concave, it follows that every operator connection is jointly concave.Example 20.
The functionf(x)=2x/(1+x) is operator monotone on R+ according to Corollary 12. This function gives rise to theharmonic mean: (35)A!B=2A-1+B-1-1,A,B>0.Moreover, A!B is the largest positive operator X such that(36)2A002B⩾XXXX(see [3, Theorem  I.3]).Example 21.
For eachα∈(0,1), the function gα(x)=xα is operator monotone on R+ by the Löwner-Heinz inequality (Theorem 3). Each gα corresponds to the α-weighted geometric mean:(37)A#αB=A1/2A-1/2BA-1/2αA1/2,A,B>0.Let us have a close look for the geometric mean #1/2, denoted briefly by #. In the literature, there are various characterizations of the geometric mean. This mean was firstly defined by Pusz and Woronowicz [25]:(38)A#B=max⁡T⩾0:Tx,y⩽A1/2xB1/2y∀x,y∈H,A,B⩾0.This definition coincides with the following formula given by Ando [3]:(39)A#B=A1/2A-1/2BA-1/21/2A1/2,A,B>0.Alternatively, the geometric mean of A,B>0 is the common limit of the following iterative process (see [26]):(40)A0=A,B0=B,An=An-1∇Bn-1,Bn=An-1!Bn-1.It was also pointed out in [14] that the geometric mean of A,B>0 is the unique positive solution to theRiccati equation:(41)XA-1X=B.Moreover, the geometric mean A#B is the largest self-adjoint operator X for which(42)AXXB⩾0(see [3, Theorem  I.2]). See more information of the geometric mean in [6, Section  4.1] and [10, Section  5.1].Example 22.
For eachp∈[-1,1] and α∈[0,1], consider the operator monotone function (see e.g., [5]):(43)fp,αx=1-α+αxp1/p,x⩾0.When p=0, it is understood that we take limit as p tends to 0 and, by L’Hôspital’s rule, (44)f0,αx=xα.Each function fp,α gives rise to a unique operator mean, namely, thequasiarithmetic power mean#p,α withexponentp andweightα as follows:(45)A#p,αB=1-αAp+αBp1/p,A,B⩾0.The family of quasiarithmetic power means includes the weighted arithmetic means (the case p=1), the weighted harmonic means (the case p=-1), and the weighted geometric means (the case p=0) as special cases.Example 23.
Letr∈[-1,1] and consider the operator monotone function (see [27])(46)grx=3r-13r+1x3r+1/2-1x3r-1/2-1,x⩾0.Since each gr is symmetric and normalized, it associates to a unique symmetric operator mean. Note that the family of these means include the arithmetic mean, the geometric mean, and the harmonic mean. The cases r=1/3 and r=-1/3 are known as thelogarithmic mean and its dual.Example 24.
For eachα∈[0,1/2], the function(47)hαx=xα+x1-α2belongs to OM(R+). Each hα gives rise to an operator mean, called theHeinz mean. See more information in [28].Example 25.
For eachp∈[-2,2], consider the operator monotone function(48)fpx=px-1xp-11/1-p,where the case p=1 is understood as(49)f1x=limp→1⁡fpx=1exx/x-1.The case p=1 gives rise to the operator mean, called theidentric mean. The case p=0 is associated to the logarithmic mean.Example 26.
The functionf(t)=log⁡t is operator monotone on (0,∞); see, for example, [5]. Fujii and Kamei [29] introduced therelative operator entropy to be the operator connection associated with this function by (33) as follows:(50)SA∣B=A1/2log⁡A-1/2BA-1/2A1/2,A,B>0.Note that SI∣A=S(A) for any A>0. The relative operator entropy satisfies attractive entropy-like properties in physical literature (see also [30]).
The relative operator entropy is closely related to the Karcher mean theory. TheKarcher mean or theweighted least-squares mean of the tuple A=(A1,A2,…,An) of strictly positive operators with weights w=(w1,w2,…,wn) of positive real numbers such that ∑wi=1 is geometrically defined to be(51)arg⁡minX>0⁡∑i=1nwiδ2X,Ai.Here, δ is a Riemannian metric given by(52)δA,B=log⁡A-1/2BA-1/2,A,B>0.It turns out that the Karcher mean (51) is the unique positive definite solution to the equation(53)∑i=1nSX∣Ai=0.See more information about Karcher mean theory in [31–34] and references therein.



---

*Source: 649839-2015-11-11.xml*