Reflexivity of the commutant and local commutants of an algebraic operator

We show that an algebraic operator on a complex Banach space has reflexive commutant if and only if each zero of the minimal polynomial of the operator is simple. Further, for any operator, the local commutant at an eigenvector is reflexive. On the other hand, for an algebraic operator whose minimal polynomial has at least one zero that is not simple, the local commutant of the operator at a given vector is reflexive precisely when the vector is an eigenvector.     

A NOTE ON MAJORIZING AND CONE ABSOLUTELY SUMMING OPERATORS IN BANACH LATTICES

Abstract

Operators T which are both majorizing and cone absolutely summing on a Banach lattice E are investigated. Compactness and nuclearity of such operators are discussed and it is shown that a trace can be defined for operators belonging to a large class of these operators. Special results are derived in the case where E is a Banach function space and T a kernel operator. Finally we derive strongly measurable representations for a certain class of cone absolutely summing operators thereby clarifying work done by Dinculeanu and J.J. Uhl.     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Geometric properties of positive definite matrices cone with respect to the Thompson metric

We shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to Thompson metric.     

A generalization of the cone expansion and compression fixed point theorem and applications

The classical Krasnosel’skii fixed point theorem concerning cone compression and expansion of norm type is extended; in it the norm is replaced with some convex functional. As an application, the existence of positive solutions for the singular second-order m$m$-point boundary value problem is considered.     

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and Newton iterative solution

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive eigenvector is convergent. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues.     

Inequalities Characterizing Coisotone Cones in Euclidean Spaces

The isotone projection cone, defined by G. Isac and A. B. Németh, is a closed pointed convex cone such that the order relation defined by the cone is preserved by the projection operator onto the cone. In this paper the coisotone cone will be defined as the polar of a generating isotone projection cone. Several equivalent inequality conditions for the coisotonicity of a cone in Euclidean spaces will be given. Thanks are due to A. B. Németh who draw the author’s attention on the relation of latticial cones generated by vectors with pairwise non-accute angles with the theory of isotone cones.     

弱内向型集压缩映射的正不动点和正特征元

本文建立了弱内向型集压缩映射的范数压缩与扩张型正不动点定理，并获得关于正特征元的一些结果．     

Perron-frobenius-type theorems for tridiagonal operators

It is proved that in a large class of bounded tridiagonal operators (infinite Jacobi matrices), not necessarily positive or non-negative, positive eigenvalues exist and the eigenvector which corresponds to the greatest of them can be taken strictly positive. It is the unique positive eigenvector up to a constant multiple.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

Geometric eigenspace of the generator ofC 0-semigroup of positive operators

LetE be a Banach lattice with order continuous norm and {T(t)} _t0 be an eventually compactc ₀-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices.     

Bemerkung zu einer Arbeit von W. Wetterling über positive Operatoren

Summary Recently Wetterling proved several inclusion theorems for the positive eigenvector of a positive operator (i.e. a linear operator mapping a cone in a Banach space into itself). These results can be interpreted in terms of oscillations of vector pairs and thus fit into the general theory of the oscillation of positive operators. The new derivation leads to slightly sharper estimates.     

Generic convergence of infinite products of positive linear operators

We present several results concerning the asymptotic behavior of (random) infinite products of generic sequences of positive linear operators on an ordered Banach space. In addition to a weak ergodic theorem we also obtain convergence to an operator of the formf(·) wheref is a continuous linear functional and is a common fixed point.     

Equivalence of communication and projective boundedness properties for monotone and homogeneous functions

This work concerns the eigenvalue problem for a monotone and homogeneous self-mapping f of a finite-dimensional positive cone. A communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f, a property that yields the existence of a nonlinear eigenvalue. Using the idea of dual function, a similar result is obtained for lower eigenspaces.     

Weighted Hardy inequalities for decreasing sequences and functions

We obtain a complete characterization of the weights for which Hardy's inequality holds on the cone of non-increasing sequences. Our proofs translate immediately to the analogous inequality for non-increasing functions, thereby also completing the investigation in that direction. As an application of our results we characterize the boundedness of the Hardy-Littlewood maximal operator on Lorentz sequence spaces.     

向量优化问题有效解的稳定性

运用标量化的方法,通过锥正定真有效解的上半连续性讨论了无限维赋范空间中锥有效解的部分上半连续性,证明了锥有效解的通有稳定性.在此基础上,进一步证明,在Baire纲的意义下,绝大多数的向量优化问题至少存在一个锥正定真有效解是本质的有效解,换句话说,绝大多数的向量优化问题锥有效解是几乎下半连续的.     

Complete hyperexpansivity, subnormality and inverted boundedness conditions

Athavale introduced in [3] the notion of a completely hyperexpansive operator. In this paper some results concerning powers of completely (alternatingly) hyperexpansive operators (not necessarily bounded) are extended tok-hyperexpansive ones. A semispectral measure is associated with a subnormal contraction as well as with a completely hyperexpansive operator, and an operator version of the Levy-Khinchin representation is obtained. Passing to the Naimark dilation of the semispectral measure, such an operator is related to a positive contraction in a natural way. New characterizations of a completely hyperexpansive operator and a subnormal contraction are given. The power bounded completely hyperexpansive operators are characterized. All these are illustrated using weighted shifts.     

Tools for the eigenvalue distribution in a non-Hermitian setting

Under mild trace norm assumptions on the perturbing sequence, we extend a recent perturbation result based on a theorem by Mirsky. The analysis concerns the eigenvalue distribution and localization of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices. Some examples of application are considered, ranging from the product of Toeplitz sequences to the approximation of PDEs with given boundary conditions. A final discussion on open questions and further lines of research ends the note.     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.