Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

This paper provides various “contractivity” results for linear operators of the form where C are positive contractions on real ordered Banach spaces X . If A generates a positive contraction semigroup in Lebesgue spaces , we show (M. Pierre's result) that is a “contraction on the positive cone ”, i.e. for all provided that .  We show also that this result is not true for 1 ⩽ . We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone . We deduce from this result that, in such spaces, is a contraction on for any positive projection C with norm 1. We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base‐norm spaces X (e.g. in real spaces or in preduals of hermitian part of von Neumann algebras), we show that for all where N is the canonical half‐norm in X . For any positive contraction C on order‐unit spaces X (e.g. on the hermitian part of a algebra), we show that is a contraction on . Applications to relative operator bounds, ergodic projections and conditional expectations are given.     

Norm Retrievable Frames and Their Perturbation in Finite Dimensional Complex Hilbert Spaces

In this article, we study the property of norm retrievability of spanning vectors in a finite dimensional complex Hilbert space ?. Using the set of zero trace operators on ? and two sets of self-adjoint operators on ? denoted by 𝒮^1,0 and 𝒮^1,1, we present some equivalent conditions to the norm retrievable frames in ?. We will also show that the property of norm retrievability for n-dimensional complex Hilbert space ? with n≠2 is stable under enough small perturbation of the frame set only for phase retrievable frames.     

Blow‐up phenomena for a semilinear parabolic equation with weighted inner absorption under nonlinear boundary flux

Blow‐up phenomena for a nonlinear divergence form parabolic equation with weighted inner absorption term are investigated under nonlinear boundary flux in a bounded star‐shaped region. We assume some conditions on weight function and nonlinearities to guarantee that the solution exists globally or blows up at finite time. Moreover, by virtue of the modified differential inequality, upper and lower bounds for the blow‐up time of the solution are derived in higher dimensional spaces. Three examples are presented to illustrate applications of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

A Liouville theorem for weighted p−Laplace operator on smooth metric measure spaces

We prove that on a smooth metric measure space with m ?Bakry–Émery curvature bounded from below by ?(m ? 1)K for some constant K ≥0 (i.e., Ric_{f ,m} ≥?(m ? 1)K ), the following degenerate elliptic equation (0.1) has no nonconstant positive solution when p > 1 and constant λ _{f ,p} satisfies Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng‐Yau's method, which was used by Wang‐Zhu in 2012 to derive a same Liouville theorem when 1 < p ≤2, Ric_{f ,m} ≥?(m ? 1)K and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.     

Well‐posedness and regularity results for a 2m‐th order parabolic equation in symmetric conical domains of ℝN+1

In this paper, we use the domain decomposition method to prove well‐posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high‐order parabolic equation set in conical time‐dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.     

The pseudomonotone polar for multivalued operators

In this work, we study the pseudomonotonicity of multivalued operators from the point of view of polarity, in an analogous way as the well-known monotone polar due to Martínez-Legaz and Svaiter, and the quasimonotone polar recently introduced by Bueno and Cotrina. We show that this new polar, adapted for pseudomonotonicity, possesses analogous properties to the monotone and quasimonotone polar, among which are a characterization of pseudomonotonicity, maximality and pre-maximality. Furthermore, we characterize the notion of D-maximal pseudomonotonicity introduced by Hadjisavvas. We conclude this work studying the connections between pseudomonotonicity, variational inequality problems and upper sign-continuity.     

Precision Matrix Estimation With ROPE

It is known that the accuracy of the maximum likelihood-based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article, we propose a ridge-type operator for the precision matrix estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context. The performance of the proposed method is illustrated by a simulation study and real data applications. Computer code used in the example analyses as well as other supplementary materials for this article are available online.