ACTA SCIENTIARUM MATHEMATICARUM最新文献

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.

We study maps between positive definite or positive semidefinite cones of unital (C^*)-algebras. We describe surjective maps that preserve

1. (1)

   the norm of the quotient or product of elements;

2. (2)

   the spectrum of the quotient or product of elements;

3. (3)

   the spectral seminorm of the quotient or product of elements.

These maps relate to the Jordan (*)-isomorphisms between the specified (C^*)-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying (C^*)-algebras, the other types of surjections can be extended to a Jordan (*)-isomorphism or a Jordan (*)-isomorphism followed by 2-sided multiplication by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize “the corollary” regarding surjections between positive semidefinite cones of unital (C^*)-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital (C^*)-algebras.

For a poset ((P;le )), the quasiorders (AKA preorders) extending the poset order “(le )” form a complete lattice F, which is a filter in the lattice of all quasiorders of the set P. We prove that if the poset order “(le )” is small, then F can be generated by few elements.

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.

Let ({mathcal {M}}_{n}) be the algebra of (ntimes n) complex matrices. Complete descriptions are given of the maps on ({mathcal {M}}_{n}) leaving invariant some unitary similarity invariant functional values of the product of matrices such as the numerical radius, the pseudo spectral radius, or the condition spectral radius.

In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital (C^*)-algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the (C^*)-algebra of operators on a Hilbert space, even to general unital (C^*)-algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and Rényi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation:

where (t in (0,1]). We prove that this equation has a unique solution and define a new mean, which we denote as (G_{t}(omega ; mathbb {A})). We explore important properties of the mean (G_{t}(omega ; mathbb {A})) including the relationship with matrix power mean, and show that the mean (G_{t}(omega ; mathbb {A})) is monotone in the parameter t. Finally, we connect the mean (G_{t}(omega ; mathbb {A})) to a barycenter for the log-determinant divergence.