Yongdo Lim

Karcher means and Karcher equations of positive definite operators

The Karcher or least-squares mean has recently become an important tool for the averaging and studying of positive definite matrices. In this paper we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means Pt in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as t → 0+. We show that each of these characterizations provide important insights about the Karcher mean.

Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory

This paper is discussed with the problem of passivity based synchronization for a class of complex dynamical networks (CDNs) consisting uncertain inner coupling matrix together with successive time-varying delays via a state feedback delayed controller. Due to occurrence of uncertainties in coupling strengths, the considered CDNs take account of an uncertain inner coupling strength which is more general than the previously existing inner coupling strengths. Specifically, the uncertainties encountering in coupling terms are characterized with the aid of interval matrix approach. Also, by introducing a simple linear transformation, the corresponding error system is formulated. Then, based on the information about control delay term, two cases are considered namely, differentiable and non-differentiable. More precisely, by constructing an appropriate Lyapunov-Krasovskii functional (LKF) containing triple integral terms in respect of Kronecker product, for both the cases, some sufficient criteria are established in terms of linear matrix inequalities (LMIs) to guarantee the robust synchronization of the addressed CDNs based on passivity property. And the established criteria optimistically reduce the L 2 gain level from the disturbance to the output vector. Subsequently, the desired state feedback gain matrix is designed in terms of the solution to a convex optimization problem. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theoretical results.

Metric convexity of symmetric cones

AbstractIn this paper we introduce a general notion of a symmetric cone, valid for thefinite and infinite dimensional case, and prove that one can de duce the seminegativecurvature of the Thompson part metric in this general setting, along with standardinequalities familiar from operator theory. As a special case, we prove that everysymmetric cone from a JB-algebra satisfies a certain convexity property for theThompson part metric: the distance function between points evolving in time on twogeodesics is a convex function. This provides an affirmative answer to a question ofNeeb [22]. 1. IntroductionLet A be a unital C -algebra with identity e, and let A + be the set of positiveinvertible elements of A. It is known that A + is an open convex cone in the spaceH(A) of hermitian elements. The geometry of A + has been studied by several au-thors. One approach has been to endow A + with a natural Finsler structure and metricand use these for a substitute for the Riemannian geometry commonly considered infinite-dimensional examples. One particular focus in this geometry has been the studyof appropriate non-positive curvature properties. One prevalent notion of non-positivecurvature is a purely metric one, that of convexity of the metric. In [3], [4] and [9],Andruchow-Corach-Stojanoff and Corach-Porta-Recht haveshown the convexity of thedistance function along two distinct geodesics and its equivalence to the well-knownLoewner-Heinz inequality. In [22], Neeb established an appropriate differential geomet-ric notion of seminegative (equal non-positive) curvature for certain classes of Finslermanifolds.Our approach is somewhat different from either of the preceding. We replace thedifferential geometric structure by the structure of a symmetric space endowed with amidpoint operation and study seminegative curvature via convexity of the metric. In[16] we obtained the convexity of the metric for symmetric spaces with weaker metricassumptions than those enjoyed by the Finsler metric on A

Symmetric spaces with convex metrics

Abstract We develop the basic theory of a general class of symmetric spaces, called lineated symmetric spaces, that satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry. Our primary interest is the case that these symmetric spaces are Banach manifolds, in which case they exhibit an interesting geometric structure, and particularly in the metric case, where it is assumed the symmetric space carries a convex metric, an invariant complete metric contracting the square root function. One major result is that the distance function between points evolving over time on two geodesics is a convex function. Primary examples arise from involutive Banach-Lie groups (G,σ) admitting a polar decomposition G = P · K, where K is the subgroup fixed by σ and P is the associated symmetric set. We consider an appropriate notion of seminegative curvature for such symmetric spaces endowed with an invariant Finsler metric and prove that the corresponding length metric must be a convex metric. The preceding results provide a general framework for the interesting Finsler geometry of the space of positive Hermitian elements of a C*-algebra that has emerged in recent years.

Applications of geometric means on symmetric cones

Abstract. Let V be a Euclidean Jordan algebra, and let

Invariant metrics, contractions and nonlinear matrix equations

\Omega

Weighted means and Karcher equations of positive operators

be the corresponding symmetric cone. The geometric mean

A Birkhoff Contraction Formula with Applications to Riccati Equations

a\#b

Weighted geometric means

of two elements a and b in

Symmetric Sets With Midpoints and Algebraically Equivalent Theories

\Omega