karcher means and karcher equations of positive definite ......some ﬁnite number of positive...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETY, SERIES BVolume 1, Pages 1–22 (January 8, 2014)S 2330-0000(2014)00003-4

KARCHER MEANS AND KARCHER EQUATIONS

OF POSITIVE DEFINITE OPERATORS

JIMMIE LAWSON AND YONGDO LIM

Abstract. The Karcher or least-squares mean has recently become an impor-tant tool for the averaging and studying of positive definite matrices. In thispaper we show that this mean extends, in its general weighted form, to theinfinite-dimensional setting of positive operators on a Hilbert space and retainsmost of its attractive properties. The primary extension is via its characteri-zation as the unique solution of the corresponding Karcher equation. We alsointroduce power means Pt in the infinite-dimensional setting and show that

the Karcher mean is the strong limit of the monotonically decreasing family ofpower means as t → 0+. We show that each of these characterizations provideimportant insights about the Karcher mean.

1. Introduction

Positive definite matrices have become fundamental computational objects inmany areas of engineering, statistics, quantum information, and applied mathe-matics. They appear as “data points” in a diverse variety of settings: covariancematrices in statistics, elements of the search space in convex and semidefinite pro-gramming, kernels in machine learning, density matrices in quantum information,and diffusion tensors in medical imaging, to cite only a few. A variety of compu-tational algorithms have arisen for approximations, interpolation, filtering, estima-tion, and averaging.

The process of averaging typically involves taking some type of matrix mean forsome finite number of positive matrices of fixed dimension. Since the pioneeringpaper of Kubo and Ando [11], an extensive theory of two-variable means has sprungup for positive matrices and operators, but the n-variable case for n > 2 hasremained problematic. Once one realizes, however, that the matrix geometric meang2(A,B) = A#B := A1/2(A−1/2BA−1/2)1/2A1/2 is the metric midpoint of A andB for the trace metric on the set P of positive definite matrices of some fixeddimension (see, e.g., [4, 13]), it is natural to use an averaging technique over thismetric to extend this mean to a larger number of variables. First M. Moakher[18] and then Bhatia and Holbrook [5] suggested extending the geometric mean ton-points by taking the mean to be the unique minimizer of the sum of the squares

Received by the editors May 18, 2012 and, in revised form, November 22, 2012.2010 Mathematics Subject Classification. Primary 47B65; Secondary 47L07, 15A48.Key words and phrases. Positive operator, operator mean, Karcher equation, Karcher mean,

power mean.The work of the second author was supported by the National Research Foundation of Korea

(NRF) grant funded by the Korean government (MEST) (No. 2012-005191).

c©2014 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)

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2 JIMMIE LAWSON AND YONGDO LIM

of the distances:

gn(A1, . . . , An) = argminX>0

n∑i=1

δ2(X,Ai),

where δ(X,Ai) = ‖ logX−1/2AiX−1/2‖. This idea had been anticipated by Elie

Cartan (see, for example, section 6.1.5 of [2]), who showed among other things sucha unique minimizer exists if the points all lie in a convex ball in a Riemannian man-ifold, which is enough to deduce the existence of the least-squares mean globallyfor P. A more detailed study of Riemannian centers of mass in the setting of Rie-mannian manifolds was carried out by H. Karcher [10], whose ideas are importantto the present work.

Another approach to generalizing the geometric mean to n-variables, indepen-dent of metric notions, was suggested by Ando, Li, and Mathias [1] via a “sym-metrization procedure” and induction. The Ando-Li-Mathias paper was also impor-tant for listing, and deriving for their mean, ten desirable properties for extendedgeometric means. Moaker and Bhatia and Holbrook were able to establish a num-ber of these important properties for the least-squares mean, but the importantquestion of the monotonicity of this mean, conjectured by Bhatia and Holbrook [5],was left open. However, the authors were recently able to show [16] that all theproperties, in particular the monotonicity, were satisfied in the more general settingof weighted means for any weight ω = (w1, ...wn) of non-negative entries summingto 1:

(P1) (Consistency with scalars) gn(ω;A) = Aw11 · · ·Awn

n if the Ai’s commute;(P2) (Joint homogeneity) gn(ω; a1A1, . . . , anAn) = aw1

1 · · · awnn gn(ω;A);

(P3) (Permutation invariance)

gn(ωσ;Aσ) = g(ω;A), where ωσ = (wσ(1), . . . , wσ(n));

(P4) (Monotonicity) if Bi ≤ Ai for all 1 ≤ i ≤ n, then gn(ω;B) ≤ gn(ω;A);(P5) (Continuity) the map gn(ω; ·) is continuous;(P6) (Congruence invariance) gn(ω;M

∗AM) = M∗gn(ω;A)M for any invertible

M ;(P7) (Joint concavity) gn(ω;λA + (1 − λ)B) ≥ λgn(ω;A) + (1 − λ)gn(ω;B) for

0 ≤ λ ≤ 1;(P8) (Self-duality) gn(ω;A

−11 , . . . , A−1

n )−1 = gn(ω;A1, . . . , An);(P9) (Determinental identity) Detgn(ω;A) =

∏ni=1(DetAi)

wi ; and(P10) (AGH weighted mean inequalities)

(

n∑i=1

wiA−1i )−1 ≤ gn(ω;A) ≤

n∑i=1

wiAi.

A key ingredient in the derivation of many of these properties, the monotonicityin particular, is the fact that the trace metric on the manifold of positive definitematrices gives them the structure of a Cartan-Hadamard Riemannian manifold,in particular a manifold of non-positive curvature. This implies that equippedwith the Riemannian distance metric the manifold is a metric space of non-positivecurvature, an NPC-space for short, also called a CAT0-space, a widely studied classof metric spaces with a rich structure (see e.g., [6], [21], [12, Chapter 11]).

Since in the quantum as well as other settings, one is interested in the moregeneral case of positive bounded linear operators on an infinite-dimensional Hilbertspace, one would like to have suitable and effective averaging procedures for this

KARCHER MEANS AND KARCHER EQUATIONS 3

context also. However, the significant theory that has developed for the multivari-able least-squares mean does not readily carry over to the setting of positive opera-tors on a Hilbert space, since one has no such Riemannian structure nor NPC-metricavailable. Nevertheless a natural question arises as to whether other (non-metric)characterizations of the least-squares mean generalize to the Hilbert space setting,a generalization that continues to satisfy the preceding properties (P1)-(P8), (P10).In this paper we suggest and shall focus on an alternative characterization of theweighted least-squares mean as the unique solution of the Karcher equation

n∑i=1

wi log(X−1/2AiX

−1/2) = 0,

show that this equation also has a unique positive solution in the infinite-dimensionalsetting, and thus refer to our generalization as the Karcher mean. (In the Riemann-ian setting the Karcher equation is the condition for the vanishing of the gradientof the least-squares distance function.)

In Section 2 we recall the Thompson metric and list properties of it that will beimportant for our development. Section 3 on power means introduces an importanttool for later developments, but the fact that well-behaved power means exist forthe Hilbert operator setting is of independent interest. Lim and Palfia [17] haverecently shown in the finite-dimensional setting that the Karcher or least-squaresmean is the limit as t → 0+ of the power means Pt. We show additionally thatthey are monotonically decreasing, which allows us to deduce the existence of theirstrong limit. In Section 4 we see that this power mean limit also exists in theHilbert space setting and use this fact for our initial provisional definition of theKarcher mean. We show in Section 5 that the Karcher mean defined in this waydoes indeed satisfy the Karcher equation, and in Section 6 we establish that it isthe unique solution and present a list of the fundamental properties of the Karchermean.

Although for convenience we carry out our work in B(H), the C∗-algebra of allbounded linear operators on a Hilbert space H, our constructions only require thatwe be working in a monotone complete subalgebra of B(H), as we point out in thelast section.

2. The Thompson metric

For a Hilbert space H let B(H) be the Banach space of bounded linear opera-tors on H equipped with the operator norm, S(H) the closed subspace of boundedself-adjoint linear operators, and let P = P(H) ⊆ S(H) be the open convex coneof positive definite operators. The Banach Lie group GL(H) of bounded invertiblelinear operators (with operation composition) acts on P via congruence transfor-mations: ΓC(X) = CXC∗. For X,Y ∈ S(H), we write X ≤ Y if Y −X is positivesemidefinite, and X < Y if Y −X is positive definite. Note that X ≤ Y if and onlyif 〈x,Xx〉 ≤ 〈x, Y x〉 for all x ∈ H.

For A,B ∈ P and t ∈ R, the t-weighted geometric mean of A and B is definedby

(2.1) A#tB = A1/2(A−1/2BA−1/2)tA1/2.

The following properties for the weighted geometric mean are well known [7,11,14].


Lemma 2.1. Let A,B,C,D ∈ P and let t ∈ R. Then

(i) A#tB = A1−tBt for AB = BA;(ii) (aA)#t(bB) = a1−tbt(A#tB) for a, b > 0;(iii) (Loewner-Heinz inequality) A#tB ≤ C#tD for A ≤ C,B ≤ D and t ∈

[0, 1];(iv) M(A#tB)M∗ = (MAM∗)#t(MBM∗) for M ∈ GL(H);(v) A#tB = B#1−tA, (A#tB)−1 = A−1#tB

−1;(vi) (λA + (1 − λ)B)#t(λC + (1 − λ)D) ≥ λ(A#tC) + (1 − λ)(B#tD) for

λ, t ∈ [0, 1];(vii) ((1− t)A−1 + tB−1)−1 ≤ A#tB ≤ (1− t)A+ tB for t ∈ [0, 1];(viii) (A#tB)#s(A#uB) = A#(1−s)t+suB for any s, t, u ∈ R. As a special case,

A#s(A#tB) = A#stB.

The Thompson metric on P is defined by d(A,B) = || log(A−1/2BA−1/2)||, where||X|| denotes the operator norm of X. It is known that d is a complete metric on P

and thatd(A,B) = max{logM(B/A), logM(A/B)},

where M(B/A) = inf{α > 0 : B ≤ αA}; see [7,20,22]. We note that the Thompsonmetric (in the second form) exists on all normal cones of real Banach spaces. Forinstance,

d((s1, . . . , sn), (t1, . . . , tn)) = max1≤i≤n

∣∣∣ log siti

∣∣∣(2.2)

on Rn+, where R+ = (0,∞).

Lemma 2.2 ([3, 7, 14]). Basic properties of the Thompson metric on P include

(i) d(A,B) = d(rA, rB) = d(A−1, B−1) = d(MAM∗,MBM∗) for any r > 0and M ∈ GL(H);

(ii) d(A#B,A) = d(A#B,B) = 12d(A,B);

(iii) d(A#tB,C#tD) ≤ (1− t)d(A,C) + td(B,D), t ∈ [0, 1];(iv) d(A#tB,A#sB) = |s− t|d(A,B), s, t ∈ [0, 1].

By the triangular inequality,

d(A#tB,C#sD) ≤ (1− t)d(A,C) + td(B,D) + |t− s|d(C,D)(2.3)

for all s, t ∈ [0, 1]. Indeed,

d(A#tB,C#sD) ≤ d(A#tB,C#tD) + d(C#tD,C#sD)

≤ (1− t)d(A,C) + td(B,D) + |t− s|d(C,D).

The topology induced by the Thompson metric on P agrees with the relativeBanach space topology [22]. It will be useful to have available some basic relation-ships between the operator norm and the Thompson metric. For this purpose weintroduce the numerical radius of A ∈ B(H):

w(A) := sup{|〈x,Ax〉| : x ∈ H, ‖x‖ = 1}.

Lemma 2.3. (i) For A ∈ B(H), (1/2)||A|| ≤ w(A) ≤ ||A||; the equality w(A) =‖A‖ holds for A ≥ 0 in S(H).

(ii) For A,B ∈ S(H), 0 ≤ A, and λ, μ ∈ R, λA ≤ B ≤ μA implies w(B) ≤max{|λ|, |μ|}w(A), and so ‖B‖ ≤ 2max{|λ|, |μ|}‖A‖. If 0 ≤ B ≤ A, then‖B‖ ≤ ‖A‖.


Proof. (i) The inequality is a standard one; see, for example, [8]. For B ∈ S(H), wehave for ‖x‖ = 1: 〈Bx,Bx〉 = 〈x,B2x〉 ≤ w(B2) ≤ ‖B2‖ = ‖B‖2, the last equalitycoming from the fact B(H) is a C∗-algebra. Taking the supremum over all ‖x‖ = 1yields ‖B‖2 ≤ w(B2) ≤ ‖B2‖ = ‖B‖2, so w(B2) = ‖B2‖. Since any A ≥ 0 has a(unique) square root B ≥ 0, the second assertion follows.

(ii) For each x ∈ H, λ〈x,Ax〉 = 〈x, λAx〉 ≤ 〈x,Bx〉 ≤ 〈x, μAx〉 = μ〈x,Ax〉,so |〈x,Bx〉| ≤ max{|λ|, |μ|}|〈x,Ax〉. Taking sups over ‖x‖ = 1 yields w(B) ≤max{|λ|, |μ|}w(A). The second assertion now follows from part (i). Taking λ = 0,μ = 1 and using w(C) = ‖C‖ for C ≥ 0 from (i) yields the last inequality. �

The following non-expansive property of addition for the Thompson metric willbe useful for our purpose.

Lemma 2.4. [15, Lemma 10.1(iv)] Let Ai, Bi ∈ P, i = 1, 2, . . . , n. Then

d(

n∑i=1

Ai,

n∑i=1

Bi) ≤ max1≤i≤n

{d(Ai, Bi)}.

Proof. For n = 2, suppose that d(A1, B1) ≤ d(A2, B2) = log r. Then A2 ≤ rB2,B2 ≤ rA2, A1 ≤ rB1, B1 ≤ rA1, and thus A1+A2 ≤ rB1+rB2 = r(B1+B2), B1+B2 ≤ rA1 + rA2 = r(A1 + A2). Hence d(A1 + A2, B1 + B2) ≤ log r = d(A2, B2).The general case easily follows by induction. �

We denote by Δn the simplex of positive probability vectors in Rn, the convex

hull of the set of unit coordinate vectors. We equip Δn with the relative Thompsonmetric given by equation (2.2)(which induces the relative Euclidean topology). Amap G : Δn × Pn → P is said to be monotonic if for any ω ∈ Δn,

G(ω;A1, . . . , An) ≤ G(ω;B1, . . . , Bn)

whenever Ai ≤ Bi and Ai, Bi ∈ P for i = 1, . . . , n. A map G : Δn × Pn → P is said

to be jointly homogeneous if G(ω; a1A1, . . . , anAn) = aw11 · · · awn

n G(ω;A1, . . . , An)for all ω = (w1, . . . , wn) ∈ Δn and Ai ∈ P, ai > 0, i = 1, . . . , n.

Proposition 2.5. Let G : Δn × Pn → P be jointly homogeneous and monotonic.

Then the following contractive property for the Thompson metric is satisfied:

d(G(ω;A), G(ω;B)) ≤n∑

i=1

wid(Ai, Bi)

for all ω = (w1, . . . , wn) ∈ Δn and A = (A1, . . . , An),B = (B1, . . . , Bn) ∈ Pn.

Proof. By definition of the Thompson metric, Ai ≤ ed(Ai,Bi)Bi and Bi ≤ ed(Ai,Bi)Ai

for all i = 1, . . . , n. By joint homogeneity and monotonicity of G,

G(ω;A1, . . . , An) ≤ G(ω; ed(A1,B1)B1, . . . , ed(An,Bn)Bn)

= e∑n

i=1 wid(Ai,Bi)G(ω;B1, . . . , Bn)

and similarly G(ω;B1, . . . , Bn) ≤ e∑n

i=1 wid(Ai,Bi)G(ω;A1, . . . , An). This impliesthat d(G(ω;A), G(ω;B)) ≤

∑ni=1 wid(Ai, Bi). �


3. Power means

Power means for positive definite matrices have recently been introduced by Limand Palfia [17]. Their notion and most of their results readily extend to the settingof positive operators on a Hilbert space, as we point out in this section.

Theorem 3.1. Let A1, . . . , An ∈ P and let ω = (w1, . . . , wn) ∈ Δn. Then for eacht ∈ (0, 1], the following equation has a unique positive definite solution:

X =n∑

i=1

wi(X#tAi).(3.4)

Proof. We show that the map f : P → P defined by f(X) =∑n

i=1 wi(X#tAi) is astrict contraction with respect to the Thompson metric. Let X,Y > 0. By Lemma2.2(i),(iii) and Lemma 2.4,

d(f(X), f(Y )) ≤ max1≤i≤n

{d(wi(X#tAi), wi(Y#tAi))}

≤ max1≤i≤n

{d(X#tAi, Y#tAi)}

≤ max1≤i≤n

{(1− t)d(X,Y )} = (1− t)d(X,Y ).

Since 1− t < 1, f is a strict contraction, hence has a unique fixed point. �

Definition 3.2 (Power means). Let A = (A1, . . . , An) ∈ Pn and ω ∈ Δn. Fort ∈ (0, 1], we denote by Pt(ω;A) the unique solution of

X =n∑

i=1

wi(X#tAi).(3.5)

For t ∈ [−1, 0), we define Pt(ω;A) = P−t(ω;A−1)−1, where A

−1 = (A−11 , . . . , A−1

n ).We call Pt(ω;A) the ω-weighted power mean of order t of A1, . . . , An.

Remark 3.3. We note that P1(ω;A) = A(ω;A) :=∑n

i=1 wiAi and P−1(ω;A) =

H(ω;A) := (∑n

i=1 wiA−1i )−1, the ω-weighted arithmetic and harmonic means of

A1, . . . , An, respectively. For t ∈ [−1, 0), Pt(ω;A) is the unique positive definitesolution of

X =

[n∑

i=1

wi(X#−tAi)−1

]−1

.(3.6)

Indeed, X−1 =∑n

i=1 wi(X−1#−tA

−1i ) if and only if X−1 = P−t(ω;A

−1).

Remark 3.4. Let t ∈ (0, 1]. Let f : P → P defined by f(X) =∑n

i=1 wi(X#tAi).Then by the Loewner-Heinz inequality, f is monotone: X ≤ Y implies that f(X) ≤f(Y ). By Theorem 3.1, f is a strict contraction for the Thompson metric with theleast contraction coefficient less than or equal to 1− t. By the Banach fixed pointtheorem

limk→∞

fk(X) = Pt(ω;A), X ∈ P.

Similarly, the map g(X) =[∑n

i=1 wi(X#tAi)−1

]−1is a strict contraction for the

Thompson metric and limk→∞ gk(X) = P−t(ω;A), for any X ∈ P.


Proposition 3.5. Let ω, μ ∈ Δn and let A,B ∈ Pn. Then for t, s ∈ (0, 1],

d(Pt(ω;A), Ps(μ;B)) ≤|s− t|

max{s, t}Δ(A) +1

max{s, t}d(ω, μ) + max1≤i≤n

d(Ai, Bi),

where Δ(A) = max1≤i,j≤n{d(Ai, Aj)} denotes the d-diameter of A = (A1, . . . , An).In particular, the map P : (0, 1]×Δn × Pn → P defined by P (t, ω,A) = Pt(ω;A) iscontinuous with respect to the Thompson metric on P.

Proof. Let ω = (w1, . . . , wn), μ = (u1, . . . , un) ∈ Δn and A = (A1, . . . , An),B =(B1, . . . , Bn) ∈ Pn.

Step 1. d(Pt(ω;A), Aj) ≤ Δ(A) for all j = 1, . . . , n.Let X = Pt(ω;A). By Lemmas 2.2 and 2.4,

d(X,Aj) = d(

n∑i=1

wi(X#tAi), Aj) = d(

n∑i=1

wi(X#tAi),

n∑i=1

wiAj)

≤ max1≤i≤n

d(X#tAi, Aj#tAj) ≤ max1≤i≤n

[(1− t)d(X,Aj) + td(Ai, Aj)]

≤ (1− t)d(X,Aj) + tΔ(A)

and thus d(X,Aj) ≤ Δ(A) for all j = 1, . . . , n.

Step 2. d(Pt(ω;A), Ps(ω;A)) ≤ s−ts Δ(A) for 0 < t ≤ s ≤ 1.

Let X = Pt(ω;A) and Y = Ps(ω;A). By Lemma 2.4 and Step 1,

d(Y,X) = d(n∑

i=1

wi(Y#sAi),n∑

i=1

wi(X#tAi)) ≤ max1≤i≤n

d(Y#sAi, X#tAi)

(2.3)≤ max

1≤i≤n[(1− s)d(Y,X) + |s− t|d(X,Ai)]

≤ (1− s)d(X,Y ) + (s− t)Δ(A),

which implies that d(X,Y ) ≤ s−ts Δ(A).

Step 3. d(Pt(ω;A), Pt(μ;A)) ≤ 1t d(ω, μ).

Let X = Pt(ω;A) and Y = Pt(μ;A). Then

d(X,Y ) = d(

n∑i=1

wi(X#tAi),

n∑i=1

ui(Y#tAi))

= d(n∑

i=1

X#t(w1/ti Ai),

n∑i=1

Y#t(u1/ti Ai))

≤ max1≤i≤n

d(X#t(w1/ti Ai), Y#t(u

1/ti Ai))

≤ max1≤i≤n

[(1− t)d(X,Y ) + td(w

1/ti Ai, u

1/ti Ai)

]= (1− t)d(X,Y ) + max

1≤i≤n| logwi − log ui| = (1− t)d(X,Y ) + d(ω, μ).


Step 4. d(Pt(ω;A), Pt(ω;B)) ≤ max1≤i≤n d(Ai, Bi).Let X = Pt(ω;A) and Y = Pt(μ;B). Then from Lemmas 2.2(i),(iii) and 2.4

d(X,Y ) = d(n∑

i=1

wi(X#tAi),n∑

i=1

wi(Y#tBi))

≤ max1≤i≤n

d(X#tAi, Y#tBi) ≤ max1≤i≤n

[(1− t)d(X,Y ) + td(Ai, Bi)]

= (1− t)d(X,Y ) + t max1≤i≤n

d(Ai, Bi).

By the triangular inequality and Steps 2-4,

d(Pt(ω;A), Ps(μ;B)) ≤ d(Pt(ω;A), Ps(ω;A)) + d(Ps(ω;A), Ps(μ;A))

+d(Ps(μ;A), Ps(μ;B))

≤ s− t

sΔ(A) +

1

sd(ω, μ) + max

1≤i≤nd(Ai, Bi)

for 0 < t ≤ s ≤ 1. �

For A = (A1, . . . , An) ∈ Pn, M ∈ GL(H), a = (a1, . . . , an) ∈ (0,∞)n, ω =(w1, . . . , wn) ∈ Δn, and for a permutation σ on n-letters, we set

MAM∗ = (MA1M∗, . . . ,MAnM

∗), Aσ = (Aσ(1), . . . , Aσσ(n)),

A(k) = (A, . . . ,A︸︷︷︸

k

) ∈ Pnk, ω(k) =

1

k(ω, . . . , ω︸︷︷︸

k

) ∈ Δnk,

at = (at1, . . . , atn), ω � a =

1∑ni=1 wiai

(w1a1, . . . , wnan) ∈ Δn,

ω =1

1− wn(w1, . . . , wn−1) ∈ Δn−1, a · A = (a1A1, . . . , anAn).

We list some basic properties of Pt(ω;A).

Proposition 3.6. Let A = (A1, . . . , An),B = (B1, . . . , Bn) ∈ Pn, ω ∈ Δn, a =(a1, . . . , an) ∈ (0,∞)n and let s, t ∈ [−1, 1] \ {0}.

(1) Pt(ω;A) = (∑n

i=1 wiAti)

1t if the Ai’s commute;

(2) Pt(ω; a · A) = (∑n

i=1 wiati)

1t Pt(ω � at;A);

(3) Pt(ωσ;Aσ) = Pt(ω;A) for any permutation σ;(4) Pt(ω;A) ≤ Pt(ω;B) if Ai ≤ Bi for all i = 1, 2, . . . , n;(5) d(Pt(ω;A), Pt(ω;B)) ≤ max1≤i≤n{d(Ai, Bi)};(6) (1− u)P|t|(ω;A) + uP|t|(ω;B) ≤ P|t|(ω; (1− u)A+ uB) for any u ∈ [0, 1];(7) Pt(ω;MAM∗) = MPt(ω;A)M

∗ for any invertible M ;(8) Pt(ω;A

−1)−1 = P−t(ω;A);(9) (

∑ni=1 wiA

−1i )−1 ≤ Pt(ω;A) ≤

∑ni=1 wiAi;

(10) Pt(ω(k);A(k)) = Pt(ω;A) for any k ∈ N;

(11) Pt(ω;A1, . . . , An−1, X) = X if and only if X = Pt(ω;A1, . . . , An−1). Inparticular, Pt(A1, . . . , An, X) = X if and only if X = Pt(A1, . . . , An);

(12) for s ∈ (0, 1], Pt(ω;X#sA1, . . . , X#sAn) = X if and only if X = Pst(ω;A);(13) if t ∈ (0, 1], then Φ(Pt(ω;A)) ≤ Pt(ω; Φ(A)) for any positive unital linear

map Φ, where Φ(A) = (Φ(A1), . . . ,Φ(An)).


Proof. Item (5) was shown in Step 4 of the previous proposition. We provide aproof of (4). The other proofs are similar to those of [17].

Suppose that Ai ≤ Bi for all i = 1, 2, . . . , n. Let t ∈ (0, 1]. Define f(X) =∑ni=1 wi(X#tAi) and g(X) =

∑ni=1 wi(X#tBi). Then Pt(ω;A) = limk→∞ fk(X)

and Pt(ω;B) = limk→∞ gk(X) for any X ∈ P, by the Banach fixed point theo-rem. By the Loewner-Heinz inequality, f(X) ≤ g(X) for all X ∈ P, and f(X) ≤f(Y ), g(X) ≤ g(Y ) whenever X ≤ Y. Let X0 > 0. Then f(X0) ≤ g(X0) andf2(X0) = f(f(X0)) ≤ g(f(X0)) ≤ g2(X0). Inductively, we have fk(X0) ≤ gk(X0)for all k ∈ N. Therefore, Pt(ω;A) = limk→∞ fk(X0) ≤ limk→∞ gk(X0) = Pt(ω;B).

Let t ∈ [−1, 0). Then A−1 ≥ B

−1 and thus P−t(ω;A−1) ≥ P−t(ω;B

−1). There-fore, Pt(ω;A) = P−t(ω;A

−1)−1 ≤ P−t(ω;B−1)−1 = Pt(ω;B). �

Remark 3.7. By Proposition 3.6(2), the power mean Pt(ω; ·) : Pn → P is homoge-neous Pt(ω; aA1, . . . , aAn) = aPt(ω;A1, . . . , An) and by (4) it is monotonic.

4. The power mean limit

In [17] Lim and Palfia have shown in the finite-dimensional setting that theKarcher or least-squares mean is the limit as t → 0+ of the (monotonically decreas-ing) family of power means Pt. We take this characterization as the launch pointfor our approach to the infinite-dimensional Karcher mean.

We recall that the strong topology on the space B(H) of bounded linear operatorsis the topology of pointwise convergence. If a net of positive semidefinite operatorsAα converges strongly to A, then the non-negative values 〈x,Aαx〉 must convergeto a non-negative 〈x,Ax〉, so the cone {A : 0 ≤ A} is strongly closed. Hence thepartial order {(A,B) ∈ S(H) × S(H) : A ≤ B} is strongly closed, since Aα ≤ Bα

and Aα → A, Bα → B in the strong topology imply Bα − Aα ≥ 0 stronglyconverges to B−A ≥ 0. We also recall the well-known fact that any monotonicallydecreasing net of self-adjoint operators that is bounded below possesses an infimumA to which it strongly converges (see, for example, Theorem 4.28(b) of [23]). Duallya monotonically increasing net that is bounded above strongly converges to itssupremum.

For G,H : Δn × Pn → P, we define G ≤ H if G(ω;A) ≤ H(ω;A) for all ω ∈ Δn

and A ∈ Pn. We note that H ≤ A, the arithmetic-harmonic mean inequality.

Theorem 4.1. Let ω ∈ Δn and A ∈ Pn. Then there exist X± ∈ P such that

limt→0±

Pt(ω;A) = X±

under the strong-operator topology. Define P0±(ω;A) = X±. Then for 0 < t ≤ s ≤1,

H = P−1 ≤ P−s ≤ P−t ≤ · · · ≤ P0− ≤ P0+ ≤ · · · ≤ Pt ≤ Ps ≤ A.

Proof. Let ω = (w1, . . . , wn) ∈ Δn and A = (A1, . . . , An) ∈ Pn.

Step 1. Pt ≤ Ps ≤ P1 = A for 0 < t ≤ s ≤ 1.Let f : P → P be defined by f(X) =

∑ni=1 wi(X#tAi). By the Banach fixed

point theorem, Pt(ω;A) = limk→∞ fk(X) for any X ∈ P. We observe from the fact


X = X#0Ai and Lemma 2.1, parts (vii) and (viii), that

f(X) =n∑

i=1

wi(X#tAi) =n∑

i=1

wi

[X# t

s(X#sAi)

]

≤n∑

i=1

wi

[(1− t

s

)X +

t

s(X#sAi)

]

=

(1− t

s

)X +

t

s

n∑i=1

wi(X#sAi).

Applying the preceding to X0 = Ps(ω;A) yields

f(X0) ≤(1− t

s

)X0 +

t

s

n∑i=1

wi(X0#sAi) =

(1− t

s

)X0 +

t

sX0 = X0.

Since f is monotonic (Remark 3.4), fk+1(X0) ≤ fk(X0) ≤ · · · ≤ f(X0) ≤ X0 forall k ∈ N. Therefore, Pt(ω;A) = limk→∞ fk(X0) ≤ X0 = Ps(ω;A).

Step 2. P−t ≤ Pt for 0 < t ≤ 1. Let X = Pt(ω;A) and let Y = P−t(ω;A).Then by definition of P−t, Y = Pt(ω;A

−1)−1 and Y −1 =∑n

i=1 wi(Y−1#tA

−1i ) =∑n

i=1 wi(Y#tAi)−1. We consider the map f(Z) =

[∑ni=1 wi(Z#tAi)

−1]−1

. Then

f(X) =

[n∑

i=1

wi(X#tAi)−1

]−1

≤n∑

i=1

wi(X#tAi) = X.

Since f is monotonic, fk(X) ≤ X for all k ∈ N. By Remark 3.4,

P−t(ω;A) = Y = limk→∞

fk(X) ≤ X = Pt(ω;A).

Step 3. H = P−1 ≤ P−s ≤ P−t for 0 < t ≤ s ≤ 1.Let X = P−s(ω;A) and Y = P−t(ω;A). By Step 1, X−1 = Ps(ω;A

−1) ≥Pt(ω;A

−1) = Y −1. Thus, X ≤ Y.

Step 4. H = P−1 ≤ P−s ≤ P−t ≤ · · · ≤ Pt ≤ Ps ≤ A, 0 < t ≤ s ≤ 1. Followsfrom Steps 1-3.

Finally the nets {Pt(ω;A)}t>0 and {P−t(ω;A)}t>0 are monotonic and boundedbetween H(ω;A) and A(ω;A). Therefore, there exist X± ∈ P such that

limt→0+

Pt(ω;A) = X+, limt→0−

Pt(ω;A) = X−

under the strong-operator topology. By Step 2, Pt ≥ P−t for all t ∈ (0, 1]. Since thepartial order on S(H) is strongly closed, their strong limits satisfy X+ ≥ X−. �Remark 4.2. The monotonically decreasing property Pt ≤ Ps,−1 ≤ t ≤ s ≤ 1, isnew, even for the finite-dimensional setting considered by Lim and Palfia [17].

Definition 4.3. We set Λ(ω;A1, . . . , An) = P0+(ω;A1, . . . , An) and call it the ω-weighted Karcher mean of A1, . . . , An.We set Λ∗(ω;A1,. . ., An)=P0−(ω;A1,. . ., An).

The basic properties of power means in Proposition 3.6 together with Theorem4.1 provide some important properties of the Karcher mean.

Theorem 4.4. (P1) (Consistency with scalars) Λ(ω;A) = Aw11 · · ·Awn

n if theAi’s commute;

(P2) (Homogeneity) Λ(ω; aA1, . . . , aAn) = aΛ(ω;A);


(P3) (Permutation invariance) Λ(ωσ;Aσ) = Λ(ω;A);(P4) (Monotonicity) if Bi ≤ Ai for all 1 ≤ i ≤ n, then Λ(ω;B) ≤ Λ(ω;A);(P5) (Continuity) d(Λ(ω;A),Λ(ω;B)) ≤ max1≤i≤n{d(Ai, Bi)} for the Thompson

metric d; in particular Λ(ω, ·) is continuous;(P6) (Invariancy) Λ(ω;M∗AM) = M∗Λ(ω;A)M for any invertible M ;(P7) (Joint concavity) Λ(ω; (1 − u)A + uB) ≥ (1 − u)Λ(ω;A) + uΛ(ω;B) for

0 ≤ u ≤ 1;(P8) (Duality) Λ(ω;A−1

1 , . . . , A−1n )−1 = Λ∗(ω;A1, . . . , An);

(P9) (AGH weighted mean inequalities) (∑n

i=1 wiA−1i )−1≤Λ(ω;A)≤

∑ni=1 wiAi.

Proof. (P1) Follows from the fact that (∑n

i=1wiAti)

1t converges to exp(

∑ni=1wi logAi)

with respect to the operator norm (see e.g. [19]) and by Proposition 3.6 and The-orem 4.1.

(P2) Let a > 0. Then

P0+(ω; aA1, . . . , aAn)x = limt→0+

Pt(ω; aA1, . . . , aAn)x = limt→0+

[aPt(ω;A1, . . . , An)x]

= a limt→0+

Pt(ω;A1, . . . , An)x = aP0+(ω;A1, . . . , An)x.

(P5). Let Xt = Pt(ω;A) and Yt = Pt(ω;B). Let α := max1≤i≤n{d(Ai, Bi)}. ByProposition 3.5, d(Xt, Yt) ≤ α for the Thompson metric d and thus Xt ≤ eαYt for0 < t ≤ 1. Since Λ(ω;A) ≤ Xt for each t, we have Λ(ω;A) ≤ eαYt. By the strongclosedness of the order and the strong convergence of Yt to Λ(ω;B) as t → 0+,Λ(ω;A) ≤ eαΛ(ω;B). Similarly Λ(ω;B) ≤ eαΛ(ω;A). It follows from definition ofthe Thompson metric that d(Λ(ω;A),Λ(ω;B)) ≤ α = max{d(Ai, Bi) : 1 ≤ i ≤ n}.

(P8). Follows from Proposition 3.6 (8),(9) and the strong continuity of inversionon bounded intervals (see Lemma 5.4).

By Proposition 3.6, Theorem 4.1 and the strong closedness of the order, theremaining parts of the proof are immediate. �

Remark 4.5. We note that P0−(ω;A) satisfies all the preceding properties exceptthe joint concavity. For a finite-dimensional Hilbert space, P0− = Λ (see [17]) andthe Karcher mean Λ(ω;A) is the unique solution of the Karcher equation

n∑i=1

wi log(X−1/2AiX

−1/2) = 0.

Then, via the Karcher equation, the Karcher mean satisfies the joint homogene-ity property ([18]) Λ(ω; a1A1, . . . , anAn) = aw1

1 · · · awnn Λ(ω;A) and the contraction

property δ(Λ(ω;A),Λ(ω;B)) ≤∑n

i=1 wiδ(Ai, Bi) with respect to the Riemannian

trace metric δ(A,B) =[∑m

i=1 log2 λi(A

−1B)] 1

2 , where the λi(X) denote the eigen-values of X. By Proposition 2.5, the contraction property holds for the Thompsonmetric. We eventually extend these results to the infinite-dimensional setting; seeTheorem 6.8.

5. The Karcher equation

Let A = (A1, . . . , An) ∈ Pn, ω = (w1, . . . , wn) ∈ Δn. We consider the followingnon-linear operator equation on P, called the Karcher equation:

n∑i=1

wi log(X−1/2AiX

−1/2) = 0.(5.7)


Note that multiplying by −1 yields the equivalent equation

n∑i=1

wi log(X1/2A−1

i X1/2) = 0,

and we pass freely between the two.In the finite-dimensional setting it is known that the least-squares mean (the

Karcher mean) satisfies the Karcher equation, indeed is the unique positive solutionof the Karcher equation (cf. Remark 4.5). Our goal in this section is to show thatthe Karcher mean we have defined in the preceding section satisfies the Karcherequation (and hence is aptly named).

Let log z denote the principal branch of the complex logarithm defined on C \(−∞, 0] defined by log z = ln |z|+ iArg z, where Arg z is the principal branch of theargument taking values in (−π, π]. By the holomorphic functional calculus logA isdefined for any bounded linear operator A with spectrum contained in C \ (−∞, 0],in particular with spectrum contained in (0,∞). We recall the following standardfact.

Lemma 5.1. For A = (A1, . . . , An) ∈ Pn, X satisfies the Karcher equation if andonly if it satisfies

n∑i=1

wi log(A−1i X) = 0.(5.8)

Proof. We note that X−1/2(X−1/2AiX−1/2)X1/2 = X−1Ai, so X−1/2AiX

−1/2 issimilar to X−1Ai, and hence the latter has positive spectrum and thus a logarithm.Applying inner automorphism by X−1/2 to equation (5.7) then yields the equation

(5.9)n∑

i=1

wi log(X−1Ai) = 0

and applying inner automorphism by X1/2 to equation (5.9) yields equation (5.7).Hence X satisfies equation (5.7) if and only if (5.9). Now multiplying equation (5.9)by −1 yields (5.8) and vice versa; hence the two have the same solution sets, andthus so do (5.7) and (5.8). �

Lemma 5.2. Operator multiplication is strongly jointly continuous when restrictedto any bounded subset.

Proof. If in the set of all operators with norm bounded by M , Aα → A stronglyand Bα → B strongly, then

‖(AαBα−AB)x‖≤‖Aα(Bα−B)x‖+‖(Aα−A)Bx‖≤M‖(Bα−B)x‖+‖(Aα−A)Bx‖,where the last two terms go to 0 by definition of strong convergence. �

The following lemma is a special case of Theorem 3.6 from Kadison’s study ofstrongly continuous operator functions [9] on self-adjoint operators.

Lemma 5.3. Let Q be an open or closed subset of R and let f : Q → R be a

continuous bounded function. Then the corresponding operator function f is strong-operator continuous on the set SHQ of bounded self-adjoint operators on a Hilbertspace H with spectra in Q.


Lemma 5.4. The following functions are strongly continuous on [e−mI, emI] ={A ∈ S(H) : e−mI ≤ A ≤ emI}.

(i) The logarithm map A �→ logA, which is also monotonic.(ii) The power map A �→ Ar for −1 ≤ r ≤ 1.(iii) The binary weighted mean map (A,B) �→ A#tB for 0 ≤ t ≤ 1.The last two functions have image contained in [e−mI, emI].

Proof. Let A ∈ [e−mI, emI]. For r < e−m, we have rI < e−mI and so 0 ≤A − e−mI < A − rI. This shows that A − rI ∈ P, hence invertible, and hencer /∈ Spec(A). In a similar fashion one sees that r > em is not in the spectrum ofA, hence spec(A) ⊆ [e−m, em]. Since log(x) and xr are continuous and bounded onthe bounded set [e−m, em], we conclude from Lemma 5.3 that the correspondingoperator functions are strongly continuous when restricted to [e−mI, emI]. It iswell known that log is monotonic on P.

For A ∈ [e−mI, emI] and 0 ≤ r ≤ 1, e−mI ≤ (e−m)rI ≤ Ar ≤ (em)rI ≤ emIby the Loewner-Heinz inequality. Since inversion is order-reversing, [e−mI, emI] isstable under inversion, and it follows that also A−r ∈ [e−mI, emI].

For A ∈ [e−mI, emI], ‖A‖ ≤ em by Lemma 2.3(ii). Since

A#tB = A1/2(A−1/2BA−1/2)tA1/2,

0 ≤ t ≤ 1, and all partial steps in the computation are bounded in norm by e3m

(since A±1/2 ≤ em/2I) for A,B ∈ [e−mI, emI], the operation is strongly continuousby Lemma 5.2 and part (ii). Since the weighted mean is monotonic and idempotent,e−mI = e−mI#te

−m ≤ A#tB ≤ emI#temI = emI. �

Lemma 5.5. Let V ∈ S(H). Then on any ball Bm(0) = {U ∈ S(H) : ||U || ≤ m}such that ||V || < m

lim(t,U)→(0,V )

etU − I

t= V,

where the limit is taken in the strong-operator topology.

Proof. Let U ∈ Bm(0). Then∥∥∥∥etU − I

t− U

∥∥∥∥ =

∥∥∥∥∥∑∞

k=01k! (tU)k − I

t− U

∥∥∥∥∥=

∥∥∥∥∥∞∑k=1

1

k!tk−1Uk − U

∥∥∥∥∥ =

∥∥∥∥∥∞∑k=2

tk−1

k!Uk

∥∥∥∥∥≤

∞∑k=2

tk−1

k!

∥∥Uk∥∥ ≤

∞∑k=2

tk−1

k!‖U‖k

≤ ‖U‖[et‖U‖ − 1

]≤ m(etm − 1).

Let x ∈ H. Then as t → 0 and U → V strongly,∥∥∥∥etU − I

tx− V x

∥∥∥∥ ≤∥∥∥∥etU − I

tx− Ux

∥∥∥∥+ ‖Ux− V x‖

≤ m(etm − 1)‖x‖+ ‖Ux− V x‖ → 0.

�


The following shows that the Karcher mean is a solution of the Karcher equation.

Theorem 5.6. For each ω ∈ Δn and A ∈ Pn, Λ(ω;A) satisfies the Karcher equa-tion (5.7 ).

Proof. Let t ∈ (0, 1]. Let Xt = Pt(ω;A) and let X0 = Λ(ω;A) = P0+(ω;A). ByTheorem 4.1, with respect to the strong topologyXt → X0 monotonically as t → 0+

and X0 = Λ(ω;A) ≤ Xt ≤ P1(ω;A) = X1 for all t ∈ [0, 1]. Pick m such thatAi, X1, X0 ∈ [e−m/2I, em/2I] for all i. It follows that Xt ∈ [e−m/2I, em/2I] for all0 ≤ t ≤ 1.

By the order reversal of inversion [e−m/2I, em/2I] is closed under inversion sothat

X−1/2t AiX

−1/2t ≤ X

−1/2t (em/2I)X

−1/2t = em/2X−1

t ≤ emI

and similarly e−mI ≤ X−1/2t AiX

−1/2t . Thus X

−1/2t AiX

−1/2t ∈ [e−mI, emI] for all

t ∈ [0, 1] and i = 1, . . . , n. By part (ii) of Lemma 2.3 we have ‖X−1/2t AiX

−1/2t ‖ ≤

em. By Lemma 5.2 and Lemma 5.4(ii), X−1/2t AiX

−1/2t converges strongly to

X−1/20 AiX

−1/20 . By strong continuity of log on [e−mI, emI] (Lemma 5.4(i)),

Ui := log(X−1/2t AiX

−1/2t ) → Vi := log(X

−1/20 AiX

−1/20 )

for all i. By Lemma 5.5 applied to any open ball Br(0), r > em, in the strongtopology

(5.10) limt→0

(X−1/2t AiX

−1/2t )t − I

t= lim

t→0

etUi − I

t= Vi = log(X

−1/20 AiX

−1/20 )

for all i = 1, . . . , n.By definition, Xt =

∑ni=1 wi(Xt#tAi). Pre- and post-multiplying this equation

by X−1/2t and substituting from equation (2.1) for the weighted mean yields for

t > 0:

I =n∑

i=1

wi(X−1/2t AiX

−1/2t )t,

that is, 0 =∑n

i=1 wi

[(X

−1/2t AiX

−1/2t )t−I

t

]. By (5.10),

0 = limt→0+

n∑i=1

wi

[(X

−1/2t AiX

−1/2t )t − I

t

]=

n∑i=1

wi limt→0+

[(X

−1/2t AiX

−1/2t )t − I

t

]

=

n∑i=1

wi log(X−1/20 AiX

−1/20 ).

This shows that X0 = Λ(ω;A) is a solution of the Karcher equation. �

Corollary 5.7. The operator Λ∗(ω;A1, . . . , An) = P0−(ω;A1, . . . , An) also satisfiesthe Karcher equation.

Proof. By definition Λ∗(ω;A) = P0−(ω;A) = limt→0− Xt(ω;A), A = (A1, . . . , An).By strong continuity (or order-reversion) of inversion and the definition of the


negative power means, Λ∗(ω;A)−1 = limt→0+ Xt(ω;A−1) = Λ(ω;A−1). By the

preceding theorem X = Λ(ω;A−1) satisfies the Karcher equation:

n∑i=1

wi log(X−1/2(A−1

i )X−1/2) = 0.

Multiplying the equation by −1 yields

0 =n∑

i=1

wi log(X1/2(Ai)X

1/2) =n∑

i=1

wi log((X−1)−1/2Ai(X

−1)−1/2) = 0,

so Λ∗(ω;A) = X−1 satisfies the Karcher equation. �

6. Uniqueness of the Karcher mean

For ω = (w1, . . . , wn) ∈ Δn and A = (A1, . . . , An) ∈ Pn, we consider the corre-

sponding Karcher equation

(6.11)

n∑i=1

wi log(X1/2A−1

i X1/2) = 0.

We denote by K(ω;A) the set of all positive definite solutions X of the Karcherequation (6.11) and consider three important properties of this set.

Lemma 6.1. (1) K(ω; a1A1, . . . , anAn) = aw11 · · · awn

n K(ω;A) for a1, . . . , an > 0.(2) MK(ω;A)M∗ = K(ω;MAM∗) for any M ∈ GL(H).

Proof. (1) Let Y = aw11 · · · awn

n X. Then

n∑i=1

wi log(Y−1/2(aiAi)Y

−1/2)

=

n∑i=1

wi log

(ai∏n

i=1 awii

X−1/2AiX−1/2

)

=n∑

i=1

wi

[log

(ai∏n

i=1 awii

I

)+ log(X−1/2AiX

−1/2)

]

= log I +n∑

i=1

wi log(X−1/2AiX

−1/2)

=n∑

i=1

wi log(X−1/2AiX

−1/2),

where the second equality follows from the fact that log tA = log tI + logA for anyt > 0 and A > 0. Therefore, the left hand side equals 0 iff the right hand side does,which translates to

Y ∈ K(ω; a · A) = K(ω; a1A1, . . . , anAn) iff Y ∈ aw11 · · · awn

n K(ω;A).

(2) Let M ∈ GL(H) and let M = UP be the polar decomposition of M , whereU∗ = U−1 and P ∈ P. From U log(A)U−1 = log(UAU−1), one computes directlyfrom the Karcher equation as given in Lemma 5.1 that

K(ω;UAU∗) = UK(ω;A)U∗.(6.12)


Let X ∈ K(ω;A). Then∑n

i=1 wi log(A−1i X) = 0 (Lemma 5.1) implies that

0 = P−1

[n∑

i=1

wi log(A−1i X)

]P =

n∑i=1

wi log(P−1A−1

i XP )

=

n∑i=1

wi log((P−1A−1

i P−1)(PXP ))

and thus PXP ∈ K(ω;PAP ). This implies that PK(ω;A)P ⊆ K(ω;PAP ). FromK(ω;A) ⊆ P−1K(ω;PAP )P−1 ⊆ K(ω;A) we conclude that

PK(ω;A)P = K(ω;PAP ).(6.13)

By (6.12) and (6.13), MK(ω;A)M∗ = K(ω;MAM∗). �

In the proof of the next theorem we use the following Banach space version ofthe Implicit Mapping Theorem, taken from [12, Theorem 5.9] and its proof.

Theorem 6.2 (The Implicit Mapping Theorem). Let U, V be open sets in Banachspaces E, F respectively, and let f : U × V → G be a C∞-mapping, where G isalso a Banach space. Let (a, b) ∈ U × V and assume that f(a, b) = 0 and that

D2f(a, b) : F → G,

where D2f is the partial with respect to the second variable, is a Banach spaceisomorphism. Then there exist neighborhoods of U0 of a and V0 of b and a C∞-map g : U0 → V0 such that g(a) = b and for (x, y) ∈ U0 × V0, f(x, y) = 0 if andonly if y = g(x).

The Loewner order intervals are defined by [A,B] := {X ∈ S(H) : A ≤ X ≤ B}and (A,B) := {X : A < X < B}.

Theorem 6.3. Let ω ∈ Δn. Then there exists εω > 1 such that for any A ∈ P,the Karcher equation (6.11) has a unique solution in [ε−1

ω A, εωA]. Furthermore, theKarcher mean Λ(ω; ·) is C∞ on a neighborhood of the diagonal in P

n.

Proof. We consider the map Fω : Pn × P → S(H) by

Fω(A1, . . . , An, X) =n∑

i=1

wi log(X1/2A−1

i X1/2),

and for A = (A1, . . . , An), FωA(X) = Fω(A1, . . . , An, X). Then Fω

A: P → S(H) is

C∞, and FωA(X) = 0 if and only if X ∈ K(ω;A).

We next employ the elementary differential calculus for open subsets of Banachspaces, as it appears, for example, in [12, Chapter I,§3-5]. The derivative of Fω

Aat

X ∈ P is a bounded linear map from the tangent space at X to the tangent space atFωA(X), which may be considered as a linear map DFω

A(X) : S(H) → S(H) since

the tangent bundles of P and S(H) are trivial with fiber S(H). Its action is givenby

DFωA(X)(Y ) = lim

t→0

FωA(X + tY )− Fω

A(X)

t.

Let I = (I, . . . , I) ∈ Pn. Then FωI(X) =

∑ni=1 wi logX = logX, so Fω

I(I) = 0 and

DFωI(I) = idS(H) (which corresponds to and follows from the well-known fact that


the exponential mapping has derivative the identity at 0). Since Fω : Pn × P →S(H) satisfies Fω(I, ·) = Fω

I(·), we conclude that the partial derivative of Fω in

the second variable, that is, the P-variable, satisfies

D2Fω(I, I) = DFω

I(I) = idS(H).

Since D2Fω(I, I) is the identity map, in particular a Banach space isomorphism,

from S(H) to S(H), it follows directly from the Implicit Mapping Theorem (The-orem 6.2) that there exist open neighborhoods UI of I in P

n and VI of I in P anda C∞-mapping g : UI → VI such that g(I) = I and Fω(A, X) = 0 if and only ifX = g(A) for all A ∈ UI, X ∈ VI .

Pick ε > 1 such that [ε−1I, εI]n ⊂ UI and [ε−1I, εI] ⊂ VI . (This is possible sinceit is easily seen that [ε−1I, εI] is the closed ε-ball in the Thompson metric.) Thenfor any A ∈ [ε−1I, εI]n, by the monotonicity and idempotency of Λ,

ε−1I = Λ(ω; ε−1I, . . . , ε−1I) ≤ Λ(ω;A) ≤ Λ(ω; εI, . . . , εI) = εI,

and thus Λ(ω;A) ∈ VI .Since we have shown that the Karcher mean always satisfies the Karcher equa-

tion, we have Fω(A,Λ(ω;A)) = 0, and hence by the preceding two paragraphsΛ(ω;A)=g(A). Thus the Karcher mean is C∞ on the open neighborhood (ε−1I, εI)n

of I and A ∈ [ε−1I, εI]n has a unique solution to the Karcher equation in [ε−1I, εI],namely the Karcher mean Λ(ω;A).

Next, let (A1, . . . , An) ∈ [ε−1A, εA]n. Set Bi = A−1/2AiA−1/2, i = 1, . . . , n. Then

Bi ∈ [ε−1I, εI] for all i. By the preceding paragraph, X = Λ(ω;B) is the uniquesolution to the Karcher equation Fω

B(X) = 0 that lies in VI . By Lemma 6.1 and

Theorem 4.4

K(ω;A1, . . . , An) ∩ [ε−1A, εA] = K(ω;A1/2B1A1/2, . . . , A1/2BnA

1/2) ∩ [ε−1A, εA]

= A1/2(K(ω;B1, . . . , Bn) ∩ [ε−1I, εI]

)A1/2

= A1/2{Λ(ω;B1, . . . , Bn)}A1/2

= {Λ(ω;A1, . . . , An)}.

This shows that the Karcher equation has the unique solution Λ(ω;A1, . . . , An) on[ε−1A, εA] and is C∞ on the open neighborhood (ε−1A, εA)n, since

Λ(ω;A1, . . . , An) = A1/2Λ(ω;A−1/2A1A−1/2, . . . , A−1/2AnA

−1/2)A1/2.

�

Theorem 6.4. The following conditions are equivalent and are all satisfied by Λ.

(i) Λ : Δn × Pn → P is jointly homogeneous;(ii) Λ : Δn × Pn → P is contractive for the Thompson metric;(iii) the equation

X = Λ(ω;X#tA1, . . . , X#tAn)

has a unique solution in P for all t ∈ (0, 1), ω ∈ Δn and A ∈ Pn;(iv) K(ω;A) = {Λ(ω;A)} for all ω ∈ Δn and A ∈ P

n.

Furthermore, X = Λ(ω;X#tA1, . . . , X#tAn) if and only if X = Λ(ω;A1, . . . , An),and X = Λ(ω;A1, . . . , An−1, X) if and only if X = Λ(ω;A1, . . . , An−1) for allt ∈ [0, 1] and ω ∈ Δn and Ai ∈ P, i = 1, . . . , n.


Proof. We first show the Λ satisfies condition (iii). Fix ω ∈ Δn and A=(A1, . . . , An).Define ft : P → P defined by ft(X) = Λ(ω;X#tA1, . . . , X#tAn). Then

d(ft(X), ft(Y )) = d(Λ(ω;X#tA1, . . . , X#tAn),Λ(ω;Y#tA1, . . . , Y#tAn)

≤ d(X#tAj , Y#tAj) := max{d(X#tAi, Y#tAi) : 1 ≤ i ≤ n}≤ (1− t)d(X,Y )

where the first inequality follows from Theorem 4.4(P5) and the second from Lemma2.2(iii). It follows that ft is a strict contraction for the Thompson metric and hencehas a unique fixed point, which is the unique solution for the equation of (iii).

Next, we establish the equivalence between (i)-(iv).(i) implies (ii): By Proposition 2.5 and the monotonicity of Λ (Theorem 4.4).(ii) implies (iii): Let ω = (w1, . . . , wn) ∈ Δn and let A = (A1, . . . , An) ∈ P

n.For t ∈ (0, 1), it follows from Lemma 2.2(iii) and the hypothesis that the mapft : P → P defined by ft(X) = Λ(ω;X#tA1, . . . , X#tAn) is a strict contractionfor the Thompson metric and hence has a unique fixed point on P, i.e., X =Λ(ω;X#tA1, . . . , X#tAn) has a unique solution in P.

(iii) implies (iv): Let X ∈ K(ω;A1, . . . , An) and pick εω > 1 as in Theorem 6.3.Pick t ∈ (0, 1) such that

(X−1/2AiX−1/2)t ∈ [ε−1

ω I, εωI], i = 1, . . . , n.

Then∑n

i=1 wi log(X−1/2AiX

−1/2) = 0 =∑n

i=1 wi log((X−1/2AiX

−1/2)t). Clearly

Y = I is then a solution of∑n

i=1 wi log(Y−1/2(X−1/2AiX

−1/2)tY −1/2) = 0, andby the monotonicity and idempotency of Λ,

Λ(ω; (X−1/2A1X−1/2)t, . . . , (X−1/2AnX

−1/2)t)

must belong to [ε−1ω I, εωI]. By the uniqueness of the Karcher solution on [ε−1

ω I, εωI](Theorem 6.3),

I = Λ(ω; (X−1/2A1X−1/2)t, . . . , (X−1/2AnX

−1/2)t).

By invariancy under congruence transformations (property (P6) of Theorem 4.4),

X = Λ(ω;X1/2(X−1/2A1X−1/2)tX1/2, . . . , X1/2(X−1/2AnX

−1/2)tX1/2)

= Λ(ω;X#tA1, . . . , X#tAn).

Since Λ(ω;A1, . . . , An) ∈ K(ω;A) is one possibility for our original choice of X,property (iii) implies X = Λ(ω;A). This shows that K(ω;A) = {Λ(ω;A)} and thatΛ(ω;A) is the unique solution of X = Λ(ω;X#tA1, . . . , X#tAn).

(iv) implies (i): Follows from Lemma 6.1(1).Finally, suppose that one of the equivalent conditions (i)-(iv) holds true. Let

X = Λ(ω;A1, . . . , An−1). Then∑n−1

i=1wi

1−wnlog(X−1/2AiX

−1/2) = 0 and hence

0 =

n−1∑i=1

wi

1− wnlog(X−1/2AiX

−1/2) +wn

1− wnlog(X−1/2XX−1/2).

By (iv), X ∈ K(ω;A1, . . . , An−1, X) = {Λ(ω;A1, . . . , An−1, X)}, that is,

X = Λ(ω;A1, . . . , An−1, X).


By (ii) the map f : P → P defined by f(Z) = Λ(ω;A1, . . . , An−1, Z) is a strictcontraction for the Thompson metric and has a unique fixed point. Thus

Λ(ω;A1, . . . , An−1, Z) = Z

if and only if Z = X = Λ(ω;A1, . . . , An−1). �

We isolate as a separate theorem the main result of the preceding, a restatementof item (iv).

Theorem 6.5. For A1, . . . , An ∈ P, the Karcher mean X = Λ(ω;A1, . . . , An) isthe unique solution of the Karcher equation

0 =n∑

i=1

ωi log(X−1/2AiX

−1/2).

Remark 6.6. In light of Theorem 6.5 it is more natural to define the Karchermean Λ(ω;A1, . . . , An) to be the unique solution of the corresponding Karcherequation. That was certainly our motivation in naming it the Karcher mean fromthe beginning.

By Corollary 5.7 Λ∗(ω;A) = limt→0− Pt(ω;A) also satisfies the same Karcherequation as Λ(ω;A) and hence by the uniqueness of solution, the two are equal.This yields the following corollary.

Corollary 6.7. For a weight ω = (w1, ...wn) ∈ Δn and A = (A1, . . . , An) ∈ Pn,Λ∗(ω;A) = Λ(ω;A), and thus Λ(ω;A) = limt→0 Pt(ω;A).

We gather together our results about the fundamental properties of the Karchermean.

Theorem 6.8. For a weight ω = (w1, ...wn) ∈ Δn and A = (A1, . . . , An) ∈ Pn, the

following properties hold.

(P1) (Consistency with scalars) Λ(ω;A) = Aw11 · · ·Awn

n if the Ai’s commute;(P2) (Joint homogeneity) Λ(ω; a1A1, . . . , anAn) = aw1

1 · · · awnn Λ(ω;A);

(P3) (Permutation invariance)

Λ(ωσ;Aσ) = Λ(ω;A), where ωσ = (wσ(1), . . . , wσ(n));

(P4) (Monotonicity) if Bi ≤ Ai for all 1 ≤ i ≤ n, then Λ(ω;B) ≤ Λ(ω;A);(P5) (Continuity) Λ is contractive for the Thompson metric;(P6) (Congruence invariance) Λ(ω;M∗AM) = M∗Λ(ω;A)M for any invertible

M ;(P7) (Joint concavity) Λ(ω;λA + (1 − λ)B) ≥ λΛ(ω;A) + (1 − λ)Λ(ω;B) for

0 ≤ λ ≤ 1;(P8) (Self-duality) Λ(ω;A−1

1 , . . . , A−1n )−1 = Λ(ω;A1, . . . , An); and

(P9) (AGH weighted mean inequalities)

(

n∑i=1

wiA−1i )−1 ≤ Λ(ω;A) ≤

n∑i=1

wiAi.

Proof. By Theorem 4.4 and Theorem 6.4, Λ is jointly homogeneous and is contrac-tive for the Thompson metric. The remaining properties appear in Theorem 4.4,except that (P8) is modified in light of the preceding corollary. �


Remark 6.9. The Karcher mean is uniquely determined by congruence invariancy(P6), self-duality (P8), and the following property

(Y)n∑

i=1

wi logAi ≤ 0 implies Λ(ω;A1, . . . , An) ≤ I

for all ω ∈ Δn and (A1, . . . , An) ∈ Pn. Furthermore, Λ(ω;A1, . . . , An) ≤ I impliesthat Λ(ω;Ap

1, . . . , Apn) ≤ I for all p ≥ 1, and the following statements are equivalent;

(i)∑n

i=1 logAi ≤ 0;(ii) Λ( 1n , . . . ,

1n ;A

p1, . . . , A

pn) ≤ I for all p > 0; and

(iii) Λ(ωp;Ap1

1 , . . . , Apnn ) ≤ I for all p = (p1, . . . , pn) ∈ Rn

+, where

ωp =1∑n

i=1 p �=i(p �=1, . . . ,p �=n) ∈ Δn, p �=i :=

∏j �=i

pj .

For a finite-dimensional Hilbert space, property (Y ) and the previous characteri-zation for the Karcher mean appear in [24] and [17], respectively. The proofs areheavily dependent on property (Y) and Hansen’s inequality and are similar to thoseof [17, 24].

7. Subalgebras

For convenience and ease of presentation we have limited our considerations tothe full algebra B(H) of bounded linear operators. However, we observe that theconstructions can be carried out in large classes of subalgebras (which we assumealways to contain the identity I). For any norm-closed C∗-subalgebra A of B(H),PA = A∩P will be its open cone of positive operators, and will be closed under theoperation of taking weighted geometric means A#tB. Hence it will be closed undertaking power means Pt(ω;A), since the power mean is the limit in the Thompsonmetric of a contractive map defined from the weighted geometric means on PA, andsince the topology of the Thompson metric agrees with the relative operator normtopology. Since we have defined Λ(ω;A) to be the strong limit of the monotonicallydecreasing family {Pt : t > 0}, we need that the subalgebra is monotone complete(actually, monotone σ-complete will suffice since one can restrict to t = 1/n andobtain the same infimum). Once one has closure under the Karcher mean for thesubalgebra, then one sees readily that its properties that we have derived for the fullalgebra B(H) are inherited by the subalgebra, in particular its characterization asthe unique solution of the corresponding Karcher equation. Since the von Neumannsubalgebras are strongly closed, they in particular have Karcher means defined inthe manner of this paper and satisfying the properties derived for it.

8. Open problems

We close with three open problems:

Problem 1: Is the strong convergence limt→0 Pt = Λ of the power means to theKarcher mean actually convergence in the operator norm topology?

Problem 2: For a given weight ω, is the map (A1, . . . , An) �→ Λ(ω;A1, . . . , An)from Pn to P a C∞-map? (We established this in a neighborhood of the diagonal.)

Problem 3: For a given weight ω, is the map (A1, . . . , An) �→ Λ(ω;A1, . . . , An)from P

n to P strongly continuous on order intervals [e−mI, emI]?


Acknowledgement

The authors wish to express their gratitude to Professor Dick Kadison for helpfulinsights about the strong topology, especially for his pointing us to reference [9].The authors also thank the referee for helpful comments, particularly for pointingout a needed correction in the authors’ use of the Implicit Function Theorem in theproof of Theorem 6.3.

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Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana

70803

E-mail address: [email protected]

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea

E-mail address: [email protected]







karcher means and karcher equations of positive definite ......some ﬁnite number of positive...

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