A CHARACTERIZATION OF THE HARMONIC OPERATOR MEAN AS AN EXTENSION OF ANDO S THEOREM. Jun Ichi Fujii* and Masahiro Nakamura** Received December 12, 2005

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1 Scientiae Mathematicae Japonicae Online, e-006, A CHARACTERIZATION OF THE HARMONIC OPERATOR MEAN AS AN EXTENSION OF ANDO S THEOREM Jun Ichi Fujii* and Masahiro Nakamura** Received December, 005 Abstract. We show that the weighted harmonic operator mean is characterized as an operator mean m satisfying F AmB F AmF B for every operator monotone function F on 0, based on the numerical means. We also show the non-affine representing function f m =m of an operator mean m is an etreme point of the set of representing functions F with F f m 5 f m F. Introduction. Let us consider the arithmetic operator mean A B =A + B/ for a pair of positive invertible operators A and B acting on a Hilbert space H. Then a real function F is operator concave if F A B F A F B holds. It is known that every operator monotone function f on 0, satisfying fa fb whenever 0 A B is operator concave. The harmonic operator mean! is defined by A + B A! B = and T.Ando [, Theorem III.5] showed the contrastive result to the above: Theorem Ando. If F is positive operator monotone, then F A! B F A!F B. In this note, based on this inequality, we discuss when F A mb F A mfb holds not only for numerical means but also operator means in the sense of Kubo and Ando [5] which can be constructed as AmB = A / fa / BA / A / for a positive operator monotone function f on 0, with f =. 000 Mathematics Subject Classification. 47A64, 47A63, 6A5, 6D07. Key words and phrases. operator mean, operator monotone function, concave function.

2 8 JUN ICHI FUJII AND MASAHIRO NAKAMURA Numerical mean. Let Ma, b be a positive homogeneous mean for positive numbers a and b. According to the Kubo-Ando theory [5], the operations M a, b =Mb, a and M a, b =M/a, /b are called the transpose and adjoint for M respectively. The symbols w,# w and! w denote the arithmetic, geometric and harmonic means respectively for 0 <w<: Then w a, b = wa + wb, # w a, b =a w b w and! w a, b = w =! w,! w = w and # w =# w ab wa + wb. and these means are all symmetric for w =/, i.e., M = M. These operations and are also applied to the representing function f M =M, for M: f = f and f =f. Note that the normalized condition Ma, a =a is equivalent to f M =. By homogeneity, such means are reconstructed by the representing functions: Ma, b =af M b/a =bf M a/b. Here we assume that f M is positive, monotone-increasing and concave. Then so is fm. In fact, it is clear that fm is positive and monotone-increasing. The concavity follows from fm w + wy = w + wyf M w + wy w = w + wyf + wy y M w + wy wf M + wyf M = wfm +wfm y. y The adjoint fm is also positive and monotone-increasing, but it is not always concave as in the following eample: Eample. Put F =. Then F / =/ / and hence which is not concave in a neighborhood of. F =, Moreover the concavity of F is equivalent to Ando s type theorem: Lemma.. Let F be a positive monotone-increasing concave function on 0,. Then F is concave if and only if for all a, b > 0. F! w a, b! w F a,fb

3 CHARACTERIZATION OF HARMONIC MEAN 9 Proof. The concavity of F is written by F w+wy = F w+wy wf +wf y = w +w. F F By putting a =/ and b =/y, it is equivalent to F af b F! w a, b = F w + wy wf b+wf a =! wf a,fb. Thus the equivalence is shown. Here we restrict ourselves to the homogeneous numerical means M with the representing functions f M satisfying i f M, fm and f M are positive monotone-increasing concave functions. ii f M is normalized: f M = i.e., Ma, a =a. Note that i implies that the above means do not include trivial means: M l a, b =a and M r a, b =b. Net we consider when F Ma, b MF a,fb holds. Note that it is equivalent to F M a, b M F a,f b. In spite of the above situation, it holds for a special pair of a mean M! w and a function F. In fact, putting F = and Ma, b = ab, the geometric mean. Then F Ma, b = 4 ab = MF a,fb. But, considering the case that F is affine, we can characterize the harmonic mean in such means, which is an etension of Ando s theorem: Theorem.. A homogeneous mean M in the above sense is the harmonic one if and only if F Ma, b MF a,fb for all positive monotone-increasing concave functions F on 0, with the concave adjoint F and positive numbers a and b. Proof. It follows from the above lemma that holds for M =! w. Suppose M! w and holds. Then M w, so that there eists with M, + M, = +f M + <fm = M, +. Applying F = + / to, we have +f M = F M, M This contradiction shows M = w, that is, M =! w., + + = f M. y

4 30 JUN ICHI FUJII AND MASAHIRO NAKAMURA 3 Operator mean. Net we discuss the case of operators. The harmonic operator mean with a weight w is defined by A! w B = wa + wb and the arithmetic one with w is A w B = wa + wb for positive invertible operators A and B on a Hilbert space. In general, operator means here stand for the Kubo-Ando operator means defined by. Note that the representing function f m =m of a nontrivial operator mean m is a positive monotone-increasing concave function and so are f m and f m. Now we have a characterization of the harmonic operator mean! w: Theorem 3.. A nontrivial operator mean m is the weighted harmonic resp. arithmetic one if and only if 3 F A mb F A mfb resp. F A mb F A mfb for all positive operator monotone functions F and positive operators A and B. Finally we observe noncommutative eamples. As we state above, for commuting operators A and B, we have A # B = 4 AB = A # B, where # is the geometric operator mean A # B = A / A / BA / A /. But it does not hold in general and moreover we can give eamples: A # B A # B and C # B C # B. Recall the following formula in []: y 0 S =, P = y z 0 0 Put S =, A = S = z y imply S # P = P. z and B = P. Then S # B = P and S# P = P, and hence A # B = S # B = 4 P P = S # P = A # B. Net, put Then we have S # P = S = 5 3,C= S = P and S # P = 6 5 P, so that and B = P. C # B = S # P = 4 5 P 3 P = S # P = C # B. These eamples show the difficulty to discuss the class of functions F satisfying 3. So we discuss another related class in the net section.

5 CHARACTERIZATION OF HARMONIC MEAN 3 4 A class of functions. Finally we consider the set MF of all representing functions of operator means, that is, positive operator monotone functions f m on 0, with f m =. Note that! w and w belong to the boundary of MF, which corresponds to [3, Theorem 8] and [5, Theorem 4.5]: Theorem 4.. The weighted arithmetic means w resp. harmonic ones! w are the largest resp. smallest operator means whose representing functions satisfy f m = w. Proof. Every representing function f m is concave and differentiable, so we have f m f m + f m = f m + f m = f w for all >0, which shows m w. Therefore! w = w are the smallest since m n implies m n for all operator means m and n. Let Sm =Sf m be the set of all F MF satisfying 4 F f m f m F, which is derived from the case A = in 3: F f m B = F mb F mfb =mfb =f m F B. Then Sm is a closed conve subset of MF with the maimal etreme points w by the above theorem. Since the equality in 4 holds, we have f m itself belongs to Sm. This suggests that m occupies an etremal position in Sm. The above argument shows that S! w coincides with MF and S w ={f w 0 <w<}. In other words, by Theorem 4, the smallest class of S! w and S w is{f!w } and {f w } respectively. In particular, these means are etreme points of MF. Moreover it is valid in general, which is another variation of Ando s theorem: Theorem 4.. If f m be the non-affine representing function for an operator mean m, then it is an etreme point of Sm: f m et Sm. Proof. Let F + F / =f m for F k Sm. Then, putting y = f m for each >0, we have f m y = F + F y = F + F f mf + f m F f m = F f m + F f m F +F f m = f m f m = f m y. Therefore, the equality holds and hence F = F by the strict concavity of f m. Consequently F = F = f m, which implies f m et Sm. Moreover we conjecture that f m is a minimal function for Sm, that is, for all totally ordered path of representing functions f mr passing through f m, see [4] f m = min{f mr f mr Sm}. Though it is valid for m =! w and w, it is an open problem in general. Recall that for the power mean m r,w for r, the representing function f mr,w = w + w r /r,

6 3 JUN ICHI FUJII AND MASAHIRO NAKAMURA is operator monotone and hence the representing one of an operator mean. For a fied weight w, it is monotone increasing for r while the power operator mean Am r,w B is not always monotone increasing in the usual order for operators. For r 0, we obtain the geometric operator mean # w with a weight w: A # w B = A / A / BA / w A /. Now we can verify that the representing function f #w = w is the smallest one in the power ones in S# w. In fact, the monotonicity of power means shows w + w r /r w + w wr /wr w + w wr /wr w + w r /r for all 0 <r. This is equivalent to w + w r w/r w + w wr /r w + w wr /r w + w r w/r, which shows f #w = min{f mr,w f mr,w S# w }. References [] T.Ando: Topics on operator inequalities, Hokkaido Univ. Lecture Note, 978. [] J.I.Fujii: Arithmetico-geometric mean of operators, Math. Japon., 3978, [3] J.I.Fujii and M.Fujii: Some remarks on operator means, Math. Japon., 4979, [4] J.I.Fujii, M.Nakamura and S.-E.Takahasi: Cooper s approach to chaotic operator means, to appear in Sci. Math. Japon.. [5] F.Kubo and T.Ando: Means of positive linear operators, Math. Ann., * Department of Arts and Sciences Information Science, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka , Japan. address : fujii@cc.osaka-kyoiku.ac.jp ** Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka , Japan.