Junichi Fujii

By virtue of the operator geometric mean and the polar decomposition, we present a new Cauchy-Schwarz inequality in the framework of semi-inner product C*-modules over unital C*-algebras and discuss the equality case. We also give several additive and multiplicative type reverses of it. As an application, we present a Kantorovich type inequality on a Hilbert C*-module. (C) 2012 Elsevier Inc. All rights reserved.

The classical Jensen inequality is expressed by internally dividing points and so is non-commutative Jensen inequalities. In this paper, considering that it is expressed by externally dividing points, we shall discuss two non-commutative Jensen inequalities and their reverse, that is, one is a vector state version and the other is a Davis-Choi-Jensen type version.

In the framework of a pre-inner product C*-module over a unital C*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. Elezovic et al. [Math. Inequal. Appl., 8 (2005), no. 2, 223-231]. We apply our results to give Klamkin-Mclenaghan, Shisha-Mond and Cassels type inequalities. We also present a Gruss type inequality.

As a generalization of the Hiai-Petz geometries, we discuss two types of them where the geodesics are the quasi-arithmetic means and the quasi-geometric means, respectively. Each derivative of such a geodesic might determine a new relative operator entropy. Also in these cases, the Finsler metric can be induced by each unitarily invariant norm. If the norm is strictly convex, then the geodesic is the shortest. We also give examples of the shortest paths which are not the geodesics when the Finsler metrics are induced by the Ky Fan k-norms. (C) 2010 Elsevier Inc. All rights reserved.

Recently Hiai-Petz (2009) [10] discussed a parametrized geometry for positive definite matrices with a pull-back metric for a diffeomorphism to the Euclidean space. Though they also showed that the geodesic is a path of operator means, their interest lies mainly in metrics of the geometry. In this paper, we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a Finsler one. Also we discuss another type of geometry in Hiai and Petz (2009) [10] which is a generalization of Corach-Porta-Recht's (1993) one [3]. (C) 2009 Elsevier Inc. All rights reserved.

In this paper, by using the weighted geometric mean G[n, t] and the weighted arithmetic one A[n, t] due to Lawson-Lim for each t is an element of [0,1], we investigate n-variable versions of a complement of the Golden-Thompson-Segal type inequality due to Ando-Hiai: Let H(1), H(2), ... , H(n) be selfadjoint operators such that m <= H(i) <= M for i = 1,2, ... , n and some scalars m <= M. Then S(e(p(M-m)))-(2/p) parallel to G[n, t] (e(pH1), ... , e(pHn))(1/p) parallel to <= parallel to e(A[n,t](H1, ... , Hn)) parallel to <= S(e(p(M-m)))(2/p) parallel to G[n,t] (e(pH1), ... , e(pHn))(1/p) parallel to for all p > 0 and the both-hand sides of the inequality above converge to the middle-hand side as p down arrow 0, where S(center dot) is the Specht ratio and parallel to center dot parallel to stands for the operator norm.

We show that for an algebraic Riccati equation X*B-1X - T* X - X*T = C, its solutions are given by X = W + BT for some solution W of X*B-1X = C + T* BT. To generalize this, we give an equivalent condition for (BW*WA) ≥ 0 for given positive W A operators B and A, by which it can be regarded as Riccati inequality X*B-1X ≤ A. As an application, the harmonic mean B ! C is explicitly written even if B and C are noninvertible.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y) = Y * X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

In this note, we present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim in [J. Lawson, Y. Lim, A general framework for extending means to higher orders, preprint] by using a limiting process due to Ando-Li-Mathias in [T. Ando, C.-K. Li, R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004) 305-334]. As applications, we obtain a reverse of the weighted arithmetic-geometric mean inequality of n-operators via Kantorovich constant: For any positive integer n >= 2, let A(1), A(2),..., A(n) be positive invertible operators on a Hilbert space H such that 0 < m <= A(i) <= M for some scalars 0 < m < M. Then A[n, t](A(1),..., A(n)) <= (M + m)(2)/4Mm G[n, t](A(1),..., A(n)) for all 0 < t < 1 where A[n, t](resp. G[n, t,]) is the weighted arithmetic mean (resp. geometric mean). Moreover, we show an n-operators version of the Specht theorem. (C) 2007 Elsevier Inc. All rights reserved.

Upper bounds for the ratio and the difference between parallel sum and series of operator connections in the sense of Anderson-Duffin-Trapp are obtained, in which the Mond-Pecaric method for convex functions is applied: Let A and B be positive operators on a Hilbert space such that 0 < mI <= A, B <= MI for some scalars m < M. Then we show an upper bound of the difference of parallel sum and series : (A + B) - (A : B) <= 2 (M + m - root Mm) I. As an application, we show a noncommutative Kantorovich inequality: For positive operators A and B such that 0 < ml <= A, B <= MI, A + B/2 <= (M + m)(2)/4Mm (A(-1) + B-1/2)(-1), and moreover we show the following refinement: 2 root Mm/M + m A + B/2 <= A # B <= M + m/2 root Mm (A(-1) + B-1/2)(-1), where A # B is the geometric mean.

Reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel. Holevo introduced the quantum channel, and gave an expression for a random-coding lower bound involving an auxiliary function. Holevo, Ogawa, and Nagaoka conjectured that this auxiliary function is concave. Here we give a proof of this conjecture.

We consider continuously differentiable means, say C-1-means. As for quasi-arithmetic means Q(f)(x(1),...,x(n)), we need an assumption that f has no stationary points so that Q(f) might be continuously differentiable. Introducing quasi-weights for C-1-means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of C-1-means, we observe that a skew power mean M-t is a composition of power means if t is an integer. Copyright (C) 2006 Jun Ichi Fujii et al.

In this paper, we shall extend Bourin's theorem for unitarily invariant norm in the framework of operator theory on a Hilbert space by applying the Mond-Pecarie method for convex functions. Moreover we obtain the operator norm version. Among others, we show that if A and Z are positive operators on a Hilbert space iI such that 0 < ml <= Z <= MI for some scalars 0 < m < M, then for each alpha > 0 parallel to(AZ(P)A)(1/p)parallel to <= alpha r(ZA (2/p)) + beta(m, M, p, alpha)parallel to A parallel to(2/p) for all p > 1 for some suitable constant beta(m, M, p, alpha), where parallel to center dot parallel to is the operator norm and r(center dot) is the spectral radius.

We introduce a skew information of Lieb's typeSf,g(A,X)=Trf(A)Xg(A)X-Trf(A) g(A)X2for selfadjoint matrices A, X. We give conditions for f and g so that Sf,g is positive or negative. As another important application, we settle the problem posed by Yanagi, Furuichi and Kuriyama from quantum information theory. © 2004 Elsevier Inc. All rights reserved.

As a converse of the arithmetic and geometric mean inequality, Specht gave the ratio of the arithmetic one by the geometric one in 1960. We can reap the rich harvest of the Specht ratio in operator theory. In this paper, we shall present other characterizations of the chaotic order and the usual one associated with Kantorovich type inequalities via the Specht ratio. Among others, as an application of the grand Fururta inequality, we show that if A and B are positive operators and k greater than or equal to A greater than or equal to 1/k for some k greater than or equal to 1, then A greater than or equal to B is equivalent to S-k(2(p - 1)s)(2/s) A(p) greater than or equal to B-P holds for all p greater than or equal to 1, s greater than or equal to 1 such that p - 1 greater than or equal to 1/s, where the Specht ratio S-k(r) is defined for each r > 0 as S-k(r) = (k(r) - 1)k(r/kr-1)/ r e log k (k > 0, k not equal 1) and S-1(r) = 1. (C) 2003 Elsevier Inc. All rights reserved.

Recently, Yamazaki showed new order preserving operator inequalities on the usual order and the chaotic order by estimating the lower bound of the difference. Mond and Shisha gave an estimtate of the difference of the arithmetic one to the geometric one, as a converse of the arithmetic-geometric mean inequality. In this paper, by means of the Mond-Shisha difference, we shall put another interpretation on a characterization of the chaotic order associated with the difference by Yamazaki: If A > 0, MI greater than or equal to B greater than or equal to W > 0 and h = M/m > 1, then log A greater than or equal to log B is equivalent to A(P) + D(m(P), M-P)I greater than or equal to B-P for all p > 0, where D(m(P), M-P) = thetaM(P) + (1 - theta)m(P) - M(Ptheta)m(P(1-theta)) and theta = log(hp - 1/plogh) 1/plogh. Moreover, inspired by Yamazaki's work, we shall make an attempt to clarify distinction between the usual order and the chaotic order by using the Furuta inequality. Among others, we show the following parametrized order preserving operator inequalities associated with the difference: If A > 0 and MI greater than or equal to B greater than or equal to ml > 0, then for each delta is an element of [0, 1] A(delta) greater than or equal to B-delta if and only if A(P+delta) + 1/m(r) C(m(r+delta), Mr+delta, p+r+delta/r+delta)1 greater than or equal to Bp+delta for p, r > 0 where the case delta = 0 means the chaotic order.