Ritsuo Nakamoto

Norm inequalities equivalent to Heinz inequality

We investigate several norm inequalities equivalent to the Heinz inequality and discuss the equivalence relations among these norm inequalities. Here we shall show an elementary and simplified proof to the famous Heinz inequality.

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

Extensions of Heinz-Kato-Furuta inequality

We give an extension of Lins recent improvement of a generalized Schwarz inequality, which is based on the Heinz-Kato-Furuta inequality. As a consequence, we can sharpen the Heinz-Kato-Furuta inequality.

Concavity of the auxiliary function appearing in quantum reliability function

A sufficient condition on concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is noted. The validity of its sufficient condition is shown by some numerical computations.

ROTA'S THEOREM AND HEINZ INEQUALITIES

Abstract The Heinz inequality and the Heinz-Kato inequality have several equivalent formulations as considered in our preceding notes. Based on these, we discuss the position of Rotas theorem, which is shown to imply the Heinz inequalities.

The spherical gap of operators

Abstract We give a Horn-Li-Merino formula for the spherical gap to clarify relationships between the gap and the spherical gap for operators. Also we give a geometric interpretation of the spherical gap.

Operator inequalities of Malamud and Wielandt

Abstract We show that the Wielandt operator inequality and the Malamud one are equivalent and discuss some variations of them. From this point of view, we give also a proof of Malamuds multivariable inequality with its variations.

Remarks on concavity of the auxiliary function appearing in quantum reliability function in classical-quantum channels

This is an extension of results represented in ISIT2003. Concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven for 0 les s les 1Reliability 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.