Yuki Seo

Complementary inequalities to inequalities of Jensen and Ando based on the Mond–Pečarić method

Abstract Applying the Mond–Pecaric method to unital positive linear maps, we shall show several complementary inequalities to Jensens inequalities on positive linear maps and consequently obtain complementary inequalities to Andos inequalities associated with operator means. We shall apply them to obtain complementary estimates for the results by Ando, Aujla–Vasudeva and Fujii on Hadamard product and operator means.

A Diaz-Metcalf type inequality for positive linear maps and its applications

Abstract. We present a Diaz–Metcalf type operator inequality as a reverseCauchy–Schwarz inequality and then apply it to get the operator versions ofPo´lya–Szego¨’s,Greub–Rheinboldt’s, Kantorovich’s,Shisha–Mond’s, Schweitzer’s,Cassels’ and Klamkin–McLenaghan’s inequalities via a unified approach. WealsogivesomeoperatorGru¨sstypeinequalitiesand anoperatorOzeki–Izumino–Mori–Seo type inequality. Several applications are concluded as well. 1. IntroductionThe Cauchy–Schwarz inequality plays an essential role in mathematical in-equalities and its applications. In a semi-inner product space (H ,h·,·i) theCauchy–Schwarz inequality reads as follows|hx,yi| ≤ hx,xi 1/2 hy,yi 1/2 (x,y ∈ H ).There are interesting generalizations of the Cauchy–Schwarz inequality in var-ious frameworks, e.g. finite sums, integrals, isotone functionals, inner productspaces, C ∗ -algebras and Hilbert C ∗ -modules; see [5, 6, 7, 13, 17, 20, 9] and refer-ences therein. There are several reverses of the Cauchy–Schwarz inequality in theliterature: Diaz–Metcalf’s, Po´lya–Szego¨’s, Greub–Rheinboldt’s, Kantorovich’s,Shisha–Mond’s, Ozeki–Izumino–Mori–Seo’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities.Inspired by the work of J.B. Diaz and F.T. Metcalf [4], we present severalreverse Cauchy–Schwarz type inequalities for positive linear maps. We give aunified treatment of some reverse inequalities of the classical Cauchy–Schwarztype for positive linear maps.Throughout the paper B(H ) stands for the algebra of all bounded linear oper-ators acting on a Hilbert space H . We simply denote by α the scalar multiple αIof the identity operator I ∈ B(H ). For self-adjoint operators A,B the partially

Function order of positive operators based on the Mond–Pečarić method

We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond–Pecaric: Let A and B be positive operators on a Hilbert space H satisfying MI⩾B⩾mI>0. Let f(t) be a continuous convex function on [m,M]. If g(t) is a continuous increasing convex function on [m,M]∪Sp(A), then for a given α>0 A⩾B⩾0impliesαg(A)+βI⩾f(B), where β=maxm⩽t⩽M{f(m)+(f(M)−f(m))(t−m)/(M−m)−αg(t)}. As applications, we shall extend Kantorovich type operator inequalities by Furuta, Yamazaki and Yanagida, and present operator inequalities on the usual order and the chaotic order via Ky Fan–Furuta constant. Among others, we show the following inequality: If A⩾B>0 and MI⩾B⩾mI>0, then Mp−1mq−1Aq⩾(q−1)q−1qq(Mp−mp)q(M−m)(mMp−Mmp)q−1Aq⩾Bp holds for all p>1 and q>1 such that qmp−1⩽Mp−mpM−m⩽qMp−1.

On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices

In this paper, from the viewpoint of the Ando–Hiai inequality, we make a comparison among three geometric means: The Ando–Li–Mathias geometric mean, the Karcher mean and the chaotic geometric mean of positive definite matrices. Among others, we show complements of the -variable Ando–Hiai inequality for the Ando–Li–Mathias geometric mean by means of the Kantorovich constant.

Developed matrix inequalities via positive multilinear mappings

Abstract Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi–Davis–Jensen inequality f ( Φ ( A , B ) ) ≤ Φ ( f ( A ) , f ( B ) ) does not hold in general for a matrix convex function f and a positive multilinear mapping Φ. We prove this inequality under certain condition on f. Moreover, some Kantorovich and convexity type inequalities including positive multilinear mappings are presented.

BOUNDS FOR AN OPERATOR CONCAVE FUNCTION

Letandbe unital positive linear maps satisfying some conditions with respect to positive scalarsand �. It is shown that if a real valued function f is operator concave on an interval J, then � (f(�(A)) �(f(A))) � f(�(A)) �(f(A)) � �(f(�(A)) �(f(A))) for every selfadjoint operator A with spectrum �(A) � J. Moreover, an external version of estimates above is presented.

AN INEQUALITY FOR SOME NONNORMAL OPERATORS- EXTENSION TO NORMAL APPROXIMATE EIGENVALUES

Next we consider a generalization of Theorem 1 to normal approximate eigen-values. Here we call the {e„} satisfying (3) a sequence of unit eigenvectorscorresponding to a normal approximate eigenvalue X of A .Theorem 3. If {e„} is a sequence of unit vectors corresponding to a normalapproximate eigenvalue X of A, then

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.

Wielandt type extensions of the Heinz—Kato—Furuta inequality

The Wielandt inequality asserts that if a positive operator A on a Hilbert space H satisfies 0 < m ≤ A ≤ M for some 0 < m < M, then