TITLE

Projective metrics and contraction principles for complex cones

PUB. DATE
June 2009
SOURCE
Journal of the London Mathematical Society;Jun2009, Vol. 79 Issue 3, p719
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantages of being explicit and easier to estimate. We prove that this metric also satisfies a contraction principle similar to Birkhoffs theorem for the Hilbert metric. We are thus able to improve existing results on spectral gaps for complex matrices. Finally, we compare the contraction principles for the hyperbolic gauge and our metric on particular cones, including complexification of Birkhoff cones. It appears that the contraction principles for our metric and the hyperbolic gauge occur simultaneously on these cones. However, we get better contraction rates with our metric.
ACCESSION #
39988785

Related Articles

• THE SHAPES OF POLYHEDRA. Kannan, Ravi; Lovasz, Laszlo; Scarf, Herbert E. // Mathematics of Operations Research;May90, Vol. 15 Issue 2, p364

Defines the Banach-Mazur distance for a pair of nonempty full dimensional bodies. Matrices utilized for the computation of Banach-Mazur distance; Theorem and proof of the computations; Vectors of a polyhedra shape; Discussion on Hilbert metric.

• The extreme points of a class of analytic operator-valued functions. HU Yuanyan; YI Yali; ZHOU Zewen // Journal of Huazhong Normal University;Jun2014, Vol. 48 Issue 3, p314

Let H be a Hilbert space. â„¬ (H) denotes the Banach space of all bounded linear operators of H into H. A class of analytic operator-valued functions in three-dimensional Hilbert space is introduced, â„‘ = {f (z) : f(z) = zI - .... is analytic on the unit disk | z | â‰¤1, where the...

• Initial states and decoherence of histories. Scherer, Artur; Soklakov, Andrei N. // Journal of Mathematical Physics;Apr2005, Vol. 46 Issue 4, p042108

We study decoherence properties of arbitrarily long histories constructed from a fixed projective partition of a finite dimensional Hilbert space. We show that decoherence of such histories for all initial states that are naturally induced by the projective partition implies decoherence for...

• INTERSECTIONS OF THE LEFT AND RIGHT ESSENTIAL SPECTRA OF $2\times 2$ UPPER TRIANGULAR OPERATOR MATRICES This topic is supported by the NSF of China.. YUAN LI; XIU-HONG SUN; HONG-KE DU // Bulletin of the London Mathematical Society;Nov2004, Vol. 36 Issue 6, p811

In this paper, perturbations of the left and right essential spectra of $2\times 2$ upper triangular operator matrix $M_C$ are studied, where $M_C= \left(\begin{smallmatrix} A & C 0 & B \end{smallmatrix}\right)$ is an operator acting on the Hilbert space \${\cal...

• Generalized Mahalanobis depth in the reproducing kernel Hilbert space. Hu, Yonggang; Wang, Yong; Wu, Yi; Li, Qiang; Hou, Chenping // Statistical Papers;Aug2011, Vol. 52 Issue 3, p511

In this paper, Mahalanobis depth (MHD) in the Reproducing Kernel Hilbert Space (RKHS) is proposed. First, we extend the notion of MHD to a generalized version, i.e., the generalized MHD (GHMD), to make it suitable for the small sample with singular covariance matrix. We prove that GMHD is...

• Semi—Fredholm Spectrum and Weyl's Theorem for Operator Matrices. Xiao Hong Cao; Mao Zheng Guo; Bin Meng // Acta Mathematica Sinica;Feb2006, Vol. 22 Issue 1, p169

When A âˆˆ B(H) and B âˆˆ B(K) are given, we denote by M C an operator acting on the Hilbert space H âŠ• K of the form $$M_{C} = {\left( {\begin{array}{*{20}c} {A} & {C} \\ {0} & {B} \\ \end{array} } \right)} .$$ In this paper, first we give the necessary and sufficient condition...

• ON A J-POLAR DECOMPOSITION OF A BOUNDED OPERATOR AND MATRICES OF J-SYMMETRIC AND J-SKEW-SYMMETRIC OPERATORS. Zagorodnyuk, Sergey M. // Banach Journal of Mathematical Analysis;2010, Vol. 4 Issue 2, p11

In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space H as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decomposition shows an inner structure of a bounded operator in a Hilbert...

• A NEW HALF-DISCRETE MULHOLLAND-TYPE INEQUALITY WITH PARAMETERS. BICHENG YANG // Annals of Functional Analysis;2012, Vol. 3 Issue 1, p142

By means of weight functions and Hadamard's inequality, a new half-discrete Mulholland-type inequality with a best constant factor is given. A best extension with parameters, some equivalent forms, the operator expressions as well as some particular cases are also considered.

• Series Expansion of a Function's Expectation Matrix at the Zero Interval Length Limit. Demıralp, Metin // AIP Conference Proceedings;9/6/2007, Vol. 936 Issue 1, p155

Expectation matrix of a function serves us to evaluate the expectation value of an algebraic multiplication-by-a-function type operator over a specified subspace of a given Hilbert space. It is also frequently called transition matrix due to quantum mechanical tradition. This work considers...

Share