Regular operator mappings and multivariate geometric means

Transcription

1 arxiv: v3 [math.fa] 1 Apr 2014 Regular operator mappings and multivariate geometric means Fran Hansen March 15, 2014 Abstract We introduce the notion of regular operator mappings of several variables generalising the notion of spectral functions. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives. As a main application of the theory we consider geometric means of operator variables that extends the geometric mean of commuting operators and the geometric mean of two arbitrary positive definite matrices. We propose an updating condition that seems natural in many applications and prove that this condition, together with a few other natural axioms, uniquely defines the geometric mean for any number of operator variables. The means defined in this way are given by explicit formulas and are computationally tractable. 1 Introduction The geometric mean of two positive definite operators was introduced by Pusz and Woronowic [13], and their definition was soon put into the context of the axiomatic approach to operator means developed by Kubo and Ando [9]. Subsequently a number of authors [8, 1, 12, 2, 11, 10] have suggested several ways of defining means of operators for several variables as extensions of the geometric mean of two operators. 1

2 There is no satisfactory definition of a geometric mean of several operator variables that is both computationally tractable and satisfies a number of natural conditions put forward in the influential paper by Ando, Li, and Mathias [1]. We put the emphasis on methods to extend a geometric mean of variables to a mean of +1 variables, and in the process we challenge one of the requirements to a geometric mean put forward by Ando, Li, and Mathias. The symmetry condition of a geometric mean is mathematically very appealing, but the condition maes no sense in a number of applications. If for example positive definite matrices A 1,A 2,...,A correspond to measurements made at times t 1 < t 2 < t then there is no way of permuting the matrices since time only goes forward. It maes more sense to impose an updating condition 1) G A 1,...,A,1) = G A 1,...,A ) /) when moving from a mean G of variables to a mean G of + 1 variables. The condition corresponds to taing the geometric mean of copies of G A 1,...,A ) and one copy of the unit matrix. A variant condition would be to impose the equality 2) G A 1,...,A,1) = G A /) 1,...,A /) ) when updating from to variables. It is an easy exercise to realise that if we set G 1 A) = A, then either of the conditions 1) or 2) together with homogeneity uniquely defines the geometric mean of commuting operators. We prove that by setting G 1 A) = A and demanding homogeneity and a few more natural conditions, then either of the updating conditions 1) or 2) leads to unique but different solutions to the problem of defining a geometric mean of operators. The means defined in this way are given by explicit formulas, and they are computationally tractable. They possess all of the attractive properties associated with geometric operator means discussed in [1] with the notable exception of symmetry. If one emphasises either of the updating conditions 1) or 2) we are thus forced to abandon symmetry. Efficient averaging techniques of positive definite matrices are important in many practical applications; for example in radar imaging, medical imaging, and the analysis of financial data. 2

3 2 Regular operator mappings 2.1 Spectral functions Let BH) denote the set of bounded linear operators on a Hilbert space H. A function F : D BH) defined in a convex domain D of self-adjoint operators in BH) is called a spectral function, if it can be written on the form Fx) = fx) for some function f defined in a real interval I, where fx) is obtained by applying the functional calculus. The definition contains some hidden assumptions. The domain D should be invariant under unitary transformations and 3) Fu xu) = u Fx)u x D for every unitary transformation u on H. Furthermore, to pairs of mutually orthogonal projections p and q acting on H, the element pxp + qxq should be in D and the equality 4) Fpxp+qxq) = pfpxp)p+qfqxq)q hold for any x BH) such that pxp and qxq are in D. An operator function is a spectral function if and only if 3) and 4) are satisfied, cf. [3, 7]. The notion of spectral function is not immediately extendable to functions of several variables. However, we may consider the two properties of spectral functions noticed by C. Davis as a ind of regularity conditions, and they are readily extendable to functions of more than one variable. The notion of a regular map of two operator variables were studied by Effros and the author in [4], cf. also [6]. Definition 2.1. Let F : D BH) be a mapping of variables defined in a convex domain D BH) BH). We say that F is regular if i) The domain D is invariant under unitary transformations of H and Fu x 1 u,...,u x u) = u Fx 1,...,x )u for every x = x 1,...,x ) D and every unitary u on H. ii) Let p and q be mutually orthogonal projections acting on H and tae arbitrary -tuples x 1,...,x ) and y 1,...,y ) of operators in BH) such that the compressed tuples px 1 p,...,px p) and qy 1 q,...,qy p) 3

4 are in the domain D. Then the -tuple of diagonal bloc matrices is also in the domain D and px 1 p+qx 1 q,...,px p+qx q) Fpx 1 p+qx 1 q,...,px p+qx q) = pfpx 1 p,...,px p)p+qfqx 1 q,...,qx q)q. By choosing q as the zero projection in the second condition in the above definition we obtain Fpx 1 p...,px p) = pfpx 1 p,...,p p)p, which shows that F for any orthogonal projection p on H may be considered as a regular operator mapping where the compressed domain D p = {x 1,...,x ) F : D p BpH), BpH) x 1 01 p),...,x 01 p) ) D}. m=1 With this interpretation we may unambiguously calculate bloc matrices by the formula ) )) ) x1 0 x 0 Fx1,...,x F,..., = ) 0 0 y 1 0 y 0 Fy 1,...,y ) which is well-nown from mappings generated by the functional calculus. 2.2 Jensen s inequality for regular operator mappings Theorem 2.2. Consider the domain D = {A 1,...,A ) A 1,...,A 0} of -tuples of positive semi-definite operators on a Hilbert space H, and let F : D BH) sa be a convex regular mapping of D into self-adjoint operators acting on H. 4

5 i) Let C be a contraction on H. If F0,...,0) 0 then the inequality FC A 1 C,...,C A C) C FA 1,...,A )C holds for -tuples A 1,...,A ) in D. ii) Let X and Y be operators acting on H with X X+Y Y = 1. Then the inequality F ) X A 1 X +Y B 1 Y,...,X A X +Y B Y X FA 1,...,A )X +Y FB 1,...,B )Y holds for -tuples A 1,...,A ) and B 1,...,B ) in D. Proof. By setting T = 1 C C) 1/2 and S = 1 CC ) 1/2 we obtain that the bloc matrices ) ) C S C S U = T C and V = T C are unitary operators on H H. Furthermore, ) 1 A 0 2 U U + 1 ) ) A 0 C V V = AC SAS for any operator A BH). By using that F is a convex regular map we 5

6 obtain FC A 1 C,...,C A C) 0 ) 0 FSA 1 S,...,SA S) ) )) C = F A 1 C 0 C,..., A C 0 0 SA 1 S 0 SA S ) ) ) ) ) 1 A1 0 = F 2 U U V A1 0 V,..., A 0 2 U U V A 0 V ) ) 2 F U A1 0 U,...,U A 0 U )+ 12 ) ) ) F V A1 0 V,...,V A 0 V = 1 ) )) A1 0 A 0 2 U F,..., U + 1 ) )) V A1 0 A 0 F,..., V = 1 ) FA1,...,A ) 0 2 U U + 1 ) 0 F0,...,0) 2 V FA1,...,A ) 0 V 0 F0,...,0) 1 ) FA1,...,A ) 0 2 U U + 1 ) V FA1,...,A ) 0 V 0 0 ) C = FA 1,...,A )C 0, 0 SFA 1,...,A )S where we used convexity in the first inequality, and in the second inequality used F0,...,0) 0. The first statement now follows. In order to prove ii) we define the map GA 1,...,A ) = FA 1,...,A ) F0,...,0) A 1,...,A ) D. Unitary invariance of F implies that F0,...,0) is a multiple of the unit operator and thus commutes with all projections. Therefore G is regular and convex with G0,..., 0) = 0. We then define bloc matrices ) ) X 0 Am 0 C = and Z Y 0 m =, m = 1,..., 0 B m and notice that C Z m C = ) X A m X +Y B m Y

7 for m = 1,...,. Finally we use i) to obtain G ) X A 1 X +Y B 1 Y,...,X A X +Y B Y 0 0 ) 0 G ) ) X = A 1 X +Y B 1 Y,...,X A X +Y B Y 0 0 G0,...,0) = G C Z 1 C,...,C Z C ) ) C GZ 1,...,Z )C = C GA1,...,A ) 0 C 0 GB 1,...,B ) = ) X GA 1,...,A )X +Y GB 1,...,B )Y from which we deduce that F ) X A 1 X +Y B 1 Y,...,X A X +Y B Y = G ) X A 1 X +Y B 1 Y,...,X A X +Y B Y +F0,...,0) X GA 1,...,A )X +Y GB 1,...,B )Y +F0,...,0) = X FA 1,...,A )X +Y FB 1,...,B )Y X F0,...,0)X Y F0,...,0)Y +F0,...,0). Since as above F0,...,0) = c 1 for some real constant c we obtain X F0,...,0)X Y F0,...,0)Y +F0,...,0) = cx X +Y Y)+c 1 = 0, and the statement of the theorem follows. QED We shall for = 1,2,... consider the convex domain D + = {A 1,...,A ) A 1,...,A > 0} of positive definite and invertible operators acting on a Hilbert space H. 7

8 Proposition 2.3. Let F be a regular map of D+ into self-adjoint operators acting on H. We assume that i) F is convex ii) FtA 1,...,tA ) = tfa 1,...,A ) t > 0, A 1,...,A ) D +. Then FC A 1 C,...,C A C) = C FA 1,...,A )C for any invertible operator C on H and A 1,...,A ) D +. Proof. Assume first that C is an invertible contraction on H. Jensen s subhomogeneous inequality is only available for regular mappings defined in D. To ε > 0 we therefore consider the mapping F ε : D BH) by setting F ε A 1,...,A ) = Fε+A 1,...,ε+A ) Fε,...,ε). By unitary invariance of F we realise that Fε,...,ε) is a multiple of the unity. Therefore, F ε is regular and convex with F ε 0,...,0) = 0. We may thus use Jensen s sub-homogeneous inequality for regular mappings and obtain F ε C A 1 C,...,C A C) C F ε A 1,...,A )C, where we now restrict A 1,...,A ) to the domain D+ inequality to and rearrange the Fε+C A 1 C,...,ε+C A C) C FA 1 +ε,...,a +ε)c +Fε,...,ε) C Fε,...,ε)C. Since F is positively homogeneous the term Fε,...,ε) = εf1,...,1) is vanishing for ε 0 and we obtain 5) FC A 1 C,...,C A C) C FA 1,...,A )C for invertible C. Again using homogeneousness we obtain inequality 5) also for arbitrary invertible C. Then by repeated application of 5) we obtain FA 1,...,A ) C 1 FC A 1 C,...,C A C)C 1 FA 1,...,A ) and the statement follows. QED 8

9 3 The perspective of a regular map Definition 3.1. Let F: D+ BH) be a regular mapping. The perspective map P F is the mapping defined in the domain D+ by setting P F A 1,...,A,B) = B 1/2 FB 1/2 A 1 B 1/2,...,B 1/2 A B 1/2 )B 1/2 for positive invertible operators A 1,...,A and B acting on H. It is a small exercise to prove that the perspective P F is a regular mapping which is positively homogeneous in the sense that P F ta 1,...,tA,tB) = tp F A 1,...,A,B) for arbitrary A 1,...,A,B) D + and real numbers t > 0. The following theorem generalises a result of Effros [5, Theorem 2.2] for functions of one variable. Theorem 3.2. The perspective P F of a convex regular map F : D + BH) is convex. Proof. Consider tuples A 1,...,A ) and B 1,...,B ) in D + and tae λ [0, 1]. We define the operators and calculate that and C = λa +1 λ)b X = λ 1/2 A 1/2 C 1/2 Y = 1 λ) 1/2 B 1/2 C 1/2 X X +Y Y = C 1/2 λa C 1/2 +C 1/2 1 λ)b C 1/2 = 1 X A 1/2 A ia 1/2 X +Y B 1/2 B ib 1/2 Y = C 1/2 λ 1/2 A 1/2 A 1/2 A ia 1/2 λ1/2 A 1/2 C 1/2 +C 1/2 1 λ) 1/2 B 1/2 B 1/2 B ib 1/2 1 λ)1/2 B 1/2 C 1/2 = C 1/2 λa i +1 λ)b i )C 1/2 9

10 for i = 1,...,. We thus obtain P F λa 1 +1 λ)b 1,...,λA +1 λ)b ) = C 1/2 F C 1/2 λa 1 +1 λ)b 1 )C 1/2,..., C 1/2 λa +1 λ)b )C 1/2) C 1/2 = C 1/2 F X A 1/2 A 1A 1/2 X +Y B 1/2 B 1B 1/2 Y,..., X A 1/2 A A 1/2 X +Y B 1/2 B B 1/2 Y) C 1/2 C 1/2 X FA 1/2 A 1A 1/2,...,A 1/2 A A 1/2 )X +Y FB 1/2 B 1B 1/2,...,B 1/2 B B 1/2 )Y) C 1/2 = λa 1/2 FA 1/2 A 1A 1/2,...,A 1/2 A A 1/2 )A1/2 +1 λ)b 1/2 FB 1/2 B 1B 1/2,...,B 1/2 B B 1/2 )B1/2 = λp F A 1,...,A )+1 λ)p F B 1,...,B ), where we used Jensen s inequality for regular mappings. QED Proposition 3.3. Let F : D + BH) be a convex and positively homogeneous regular mapping. Then F is the perspective of its restriction G to D + given by GA 1,...,A ) = FA 1,...,A,1) for positive invertible operators A 1,...,A acting on H. Proof. Since F is a convex and positively homogeneous regular mapping we may apply Proposition 2.3. Then by setting C = A 1/2 we obtain A 1/2 FA 1,...,A,A )A 1/2 = FA 1/2 A 1A 1/2,...,A 1/2 A A 1/2,1). By rearranging this equation we obtain FA 1,...,A,A ) = A 1/2 GA 1/2 A 1A 1/2,...,A 1/2 A A 1/2 )A1/2 which is the statement to be proved. QED The result in the above proposition may be reformulated in the following way: The perspective P G of a convex regular mapping G: D+ BH) is the unique extension of G to a positively homogeneous convex regular mapping F : D+ BH). 10

11 4 The construction of geometric means We construct a sequence of multivariate geometric means G 1,G 2,... by the following general procedure. i) We begin by setting G 1 A) = A for each positive definite invertible operator A. ii) To each geometric mean G of variables we associate an auxiliary mapping F : D+ BH) such that a) F is a regular map, b) F is concave, c) F t 1,...,t ) = t 1 t ) 1/) for positive numbers t1,...,t. iii) We define the geometric mean G : D + BH) of +1 variables as the perspective of the auxiliary map F. G A 1,...,A ) = P F A 1,...,A ) Geometric means defined by this very general procedure are concave and positively homogeneous regular mappings by Theorem 3.2 and the preceding remars. They also satisfy 6) G A 1,...,A ) = A 1 A ) 1/ for commuting operators. Indeed, since G is the perspective of F and this map satisfies b) in condition ii), we obtain G t 1,...,t ) = ) 1/ t 1 t for positive numbers. Equality 6) then follows since G is regular. The geometric mean of two variables 7) G 2 A 1,A 2 ) = A 1/2 2 1/2 A 2 A 1 A 1/2 ) 1/2A 1/2 2 2 coincides with the geometric mean of two variables A 1 #A 2 introduced by Pusz and Woronowic. This is so since G 2 is the perspective of F 1 and F 1 A) = A 1/2. The last statement is obtained since F 1 is a regular mapping and satisfies F 1 t) = t 1/2 for positive numbers by b) in condition ii). There are many ways to associate the auxiliary map F in the above procedure, so we should not in general expect much similarity between the geometric means for different number of variables. 11

12 4.1 The inductive geometric mean We define the auxiliary mapping F : D + BH) by setting for = 1,2,... F A 1,...,A ) = G A 1,...,A ) /) Theorem 4.1. The means G constructed in section 4 have the following properties: i) G : D + BH) + is a regular map for each = 1,2,... ii) G ta 1,...,tA ) = tg A 1,...,A ) for t > 0, A 1,...,A ) D + and = 1,2,... iii) G : D+ BH) is concave for each = 1,2,... iv) G A 1,...,A,1) = G A 1,...,A ) /) for A 1,...,A ) D + and = 1,2,... Any sequence of mappings G beginning with G 1 A) = A and satisfying the above conditions coincide with the means G for = 1,2,... Proof. Each map G is for = 2,3,... the perspective of a regular map and this implies i) and ii). The assertion of concavity for G 1 is immediate. Suppose now G is concave for some. Since the map t t p is both operator monotone and operator concave for 0 p 1, we realise that the auxiliary mapping F A 1,...,A ) = G A 1,...,A ) /) is concave, and sinceg is the perspective off we then obtain by Theorem 3.2 that also G is concave. Since the first map G 1 is concave we have thus proved by induction that G is concave for all = 1,2... The last property iv) follows since G is the perspective of G /). Let finally G be a sequence of mappings satisfying i) to iv). Since each G is concave and homogeneous it follows by Proposition 3.3 that G is the perspective of its restriction G A 1,...,A,1). Because of iv) we then realise that G is the perspective of the map F A 1,...,A ) = G A 1,...,A ) /) 12

13 constructed from G. The G mappings are thus constructed by the same algorithm as the mappings G for every 2, and since G 1 = G 1 they must all coincide. QED In addition to the properties listed in the above theorem the means G enjoy a number of other properties that we list below. Theorem 4.2. The means G constructed in section 4 have the following additional properties: i) The means G are increasing in each variable for = 1,2... ii) The means G are congruence invariant. For any invertible operator C on H the identity G C A 1 C,...,C A C) = C G A 1,...,A )C holds for A 1,...,A ) D + and = 1,2,... iii) The means G are jointly homogeneous in the sense that G t 1 A 1,...,t A ) = t 1 t ) 1/ GA 1,...,A ) for scalars t 1,...,t > 0, operators A 1,...,A ) D + and = 1,2,... iv) The means G are self-dual in the sense that G A 1 1,...,A 1 ) = G A 1,...,A ) 1 for A 1,...,A ) D + and = 1,2,... v) When restricted to positive definite matrices the determinant identity holds for = 1,2... detg A 1,...,A ) = deta 1 deta ) 1/ Proof. The first property follows by the following standard argument for positive concave mappings. Consider positive definite invertible operators A m B m for m = 1,...,. By first assuming that the difference B m A m is invertible we may tae λ 0,1) and write λb m = λa m +1 λ)c m m = 1,...,, 13

14 where C m = λ1 λ) 1 B m A m ) is positive definite and invertible. By using concavity we then obtain G λb 1,...,λB ) λga 1,...,A )+1 λ)g C 1,...,C ) λga 1,...,A ). Letting λ 1 we obtain G B 1,...,B ) G A 1,...,A ) by continuity. In the general case we choose 0 < µ < 1 such that µa m < A m B m m = 1,..., and obtain G µa 1,...,µA ) G B 1,...,B ). By letting µ 1 we then obtain G A 1,...,A ) G B 1,...,B ) which shows i). Since G is concave and homogeneous we obtainii) from Proposition 2.3. Property iii) is immediate for = 1 and = 2. Suppose the property is verified for, then G t 1 A 1,...,t A,t A ) = t A 1/2 F = t A 1/2 G t1 t 1 A 1/2 A 1A 1/2,...,t t 1 A 1/2 A A 1/2 t1 t 1 A 1/2 A 1A 1/2,...,t t 1 A 1/2 A A 1/2 By using the induction assumption we obtain G t 1 A 1,...,t A,t A ) ) A 1/2 = t t 1 t1/ 1 t 1/ ) /) G A 1,...,A,A ) = t 1 t t ) 1/) G A 1,...,A,A ) ) /)A 1/2. which shows iii). Property iv) is immediate for = 1 and = 2. Suppose the property is verified for, then G A 1 1,...,A 1,A 1 = A 1/2 F = A 1/2 G ) 1/2 A A 1 1 A 1/2,...,A1/2 A 1 A1/2 A 1/2 A 1 1 A1/2,...,A1/2 A 1 A1/2 ) A 1/2 ) /) A 1/2. 14

15 By using the induction assumption we obtain G A 1 1,...,A 1 = A 1/2 G = A 1/2 G,A 1 ) 1/2 A A 1A 1/2,...,A 1/2 A A 1/2 1/2 A A 1A 1/2,...,A 1/2 A A 1/2 = G A 1,...,A,A ) 1 ) /)A 1/2 ) /) A 1/2 ) 1 which shows iv). Notice that since deta = exptr loga) for positive definite A, we have deta p = deta) p for all real exponents p. Property v) is easy to calculate for = 1 and = 2. Suppose the property is verified for. Since as above we obtain G A 1,...,A,A ) = A 1/2 G 1/2 A A 1A 1/2,...,A 1/2 A A 1/2 detg A 1,...,A,A ) = deta deta 1 deta 1 deta 1 deta = deta 1 deta deta ) 1/ which shows v). QED ) /) A 1/2 ) 1/) Theorem 4.3. The geometric means G are for = 1,2,... bounded between the symmetric harmonic and arithmetic means. That is, A A 1 G A 1,...,A ) A 1 + +A for arbitrary A 1,...,A ) D + and = 1,2,... Proof. The upper bound holds with equality for = 1. Suppose that we have verified the inequality for. Since by classical analysis X /) 1+ X 1) +1 15

16 for positive definite X, we obtain F A 1,...,A ) = GA 1,...,A ) /) GA 1,...,A ) 1) 1+ A1 + +A ) 1 = A 1 + +A By taing perspectives we now obtain G A 1,...,A,B) = P F A 1,...,A,B) = B 1/2 F B 1/2 A 1 B 1/2,...,B 1/2 A B 1/2 )B 1/2 B 1/2B 1/2 A 1 B 1/2 + +B 1/2 A B 1/2 +1 B 1/2 = A 1 + A +B which proves the upper bound by induction. We next use the upper bound to obtain G A 1 1,...,A 1 ) A A 1. By inversion we then obtain A A 1 G A 1 1,...,A 1 ) 1 = G A 1,...,A ), where we in the last equation used self-duality of the geometric mean, cf. property iv) in Theorem 4.2. QED The means studied in this section are nown in the literature as the inductive means of Sagae and Tanabe [14]. By considering the power mean A# t B = B 1/2 A 1/2 BA 1/2) t B 1/2 they established the recursive relation by setting 0 t 1 G A 1,...,A ) = G A 1,...,A )# /) A. The authors did not study the general properties of these means but established the harmonic-geometric-arithmetic mean inequality of Theorem 4.3. It is possible to prove the crucial concavity property iii) in Theorem 4.1 by induction. It can be done without the general theory of perspectives of regular operator mappings, and it only requires the properties of an operator mean of two variables as studied by Kubo and Ando [9]. However, this is a special situation that only applies to the inductive means. 16

17 4.2 Variant geometric means The inductive geometric means are uniquely specified within the general framewor discussed in this paper by choosing the updating condition 1), cf. property iv) in Theorem 4.1. We may instead construct geometric means satisfying updating condition 2) by choosing the auxiliary map F A 1,...,A ) = G A /) 1,...,A /) ) for = 1,2... It is a small exercise to realise that these means satisfy all of the properties listed in Theorem 4.1, Theorem 4.2, and Theorem 4.3 with the only exception that condition iv) in Theorem 4.1 is replaced by updating condition 2). Concavity of these means cannot be reduced to concavity of operator means of two variables but relies on the general theory of regular operator mappings and Theorem The Karcher means The Karcher mean Λ A 1,...,A ) of positive definite invertible operator variables is defined as the unique positive definite solution to the equation 8) log X 1/2 A i X 1/2) = 0, i=1 and it enjoys all of the attractive properties of an operator mean listed by Ando, Li, and Mathias, cf. [10]. The defining equation 8) immediately implies that the Karcher mean Λ : D+ BH) is a regular operator mapping, and it may therefore be understood within the general framewor discussed in this paper by choosing the auxiliary map F A 1,...,A ) = Λ A 1,...,A,1). The problem, however, is that we do not have any explicit expression of F in terms of Λ. References [1] T. Ando, C.-K. Li, and R. Mathias. Geometric means. Linear Algebra Appl., 385: ,

18 [2] R. Bhatia and J. Holbroo. Riemannian geometry and matrix geometric means. Linear Algebra and Its Applications, 413: , [3] C. Davis. A Schwarz inequality for convex operator functions. Proc. Amer. Math. Soc., 8:42 44, [4] E. Effros and F. Hansen. Non-commutative perspectives. Annals of Functional Analysis, 52):74 79, [5] E.G. Effros. A matrix convexity approach to some celebrated quantum inequalities. Proc. Natl. Acad. Sci. USA, 106: , [6] F. Hansen. Means and concave products of positive semi-definite matrices. Math. Ann., 264: , [7] F. Hansen and Pedersen G.K. Jensen s operator inequality. Bull. London Math. Soc., 35: , [8] H. Kosai. Geometric mean of several operators [9] F. Kubo and T. Ando. Means of positive linear operators. Math. Ann., 246: , [10] J. Lawson and Y. Lim. Karcher means and Karcher equations of positive definite operators. Trans. Amer. Math. Soc., Series B, 1:1 22, [11] Y. Lim and M. Pálfia. Matrix power means and the Karcher mean. Journal of Functional Analysis, 262: , [12] M. Moaher. A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix. Anal. Appl., 263): , [13] W. Pusz and S.L. Woronowicz. Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys., 8: , [14] M. Sagae and K. Tanabe. Upper and lower bounds for the arithmeticgeometric-harmonic means of positive definite matrices. Linear and Multilinear Algebra, 37: , Fran Hansen: Institute of Excellence in Higher Education, Tohou University, Japan. fran.hansen@m.tohou.ac.jp. 18