ACTA SCIENTIARUM MATHEMATICARUM最新文献

This paper investigates an important functional representation of the cone of bounded positive semidefinite operators. It is known that the representation by strength functions turns the Löwner order into the pointwise order. However, very little is known about the structure of strength functions. Our main result says that the representation behaves naturally with the infimum and supremum operations. More precisely, we show that the pointwise minimum of two strength functions (f_A) and (f_B) is a strength function if and only if the infimum of A and B exists. This complements a recent result of L. Molnár stating that the pointwise maximum of (f_A) and (f_B) exists if and only if A and B are comparable, as this latter statement is equivalent to the existence of the supremum. The cornerstone of each argument in this paper is a fact that was discovered recently, namely that the strength function of the parallel sum A : B (which is half of the harmonic mean) equals the parallel sum of the strength functions (f_A) and (f_B). We provide a new proof for this statement, and as a byproduct, in some special cases, we describe the strength function of the so-called (generalized) short.

The purpose of this paper is to extend symmetric means of multi-variable positive matrices to those of multi-variable positive operators on an infinite dimensional Hilbert space. We also consider the situations where symmetric means of operators A, B, C become linear combinations of them.

In this paper, we consider two generalized matrix equations that involve an arbitrary Kubo–Ando mean. We also study the multi-step stationary iterative method for these equations and prove the corresponding convergences.

We are concerned with log-majorization for matrices in connection with the multivariate Golden–Thompson trace inequality and the Karcher mean (i.e., a multivariate extension of the weighted geometric mean). We show an extension of Araki’s log-majorization and apply it to the (alpha )-z-Rényi divergence in quantum information. We discuss the equality cases in the multivariate trace inequality of Golden–Thompson type and in the norm inequality for the Karcher mean. The paper includes an appendix to correct the proof of the author’s old result on the equality case in the norm inequality for the weighted geometric mean.

This note offers some remarks on a norm version of the matrix arithmetic–geometric inequality.

Let J be a non trivial involutive Hermitian matrix. Consider ({mathbb {C}}^n) equipped with the indefinite inner product induced by J, ([x,y]=y^*J x) for all (x,yin {{mathbb {C}}}^n,) which endows the matrix algebra ({mathbb {C}}^{ntimes n}) with a partial order relation (le ^J) between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint (alpha )-weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.

The extended power difference mean (f_{a,b}(t):={bover a}{{t^a-1}over {t^b-1}}) ((a,bin {mathbb {R}})) is investigated in this paper. We show some Thompson metric inequalities involving (f_{a,b}) and Tsallis relative operator entropy. We also discuss the behavior of the bivariate function defined as the perspective map for (f_{a,b}). Finally, the relationship beween (f_{a,b}) and the weighted logarithmic mean is studied.