Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

On Automatic Continuity and Three Problems of “The Scottish Book” Concerning the Boundedness of Polynomial Functionals

In this paper we introduce and study the notions of isotropic mapping and essential kernel. In addition some theorems on the Borel graph and Baire mapping for polynomial operators are proved. It is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace.     

算子的本性相似

对作用于一复可分无穷维Hilbet空间(？)上的两个有界线性算子T,S,它 们本性相似和π(T),π(S)(T,S在Calkin代数中的象)相似是否等价？这是有趣的问 题.本文引入一个算子类ES((？)),并证明了对ES((？))中的算子,上面提到的两种相 似性是等价的. 此外,还证明了ES((？))在B((?))中按照算子范数拓扑稠密.     

Differentiability of a convex semigroup on Rn

It is proved that each continuous semigroup {P(t)}_t≥0 of convex operators P(t):Rⁿ→Rⁿ is continuously differentiable with respect to t. This note represents a first step towards a better understanding of semigroups formed by convex operators. We establish the differentiability of a convex semigroup in the finite dimensional case, generalizing a basic result from linear semigroup theory. Our motivation for the study of semigroups of convex operators comes from the theory of Markov decision processes. In [1] and in [2] it was shown that the maximum reward of these processes can be described by a certain nonlinear semigroup. The nonlinear operators are defined as suprema of linear operators (plus a constant), hence they are convex operators. It seems that the convexity assumption keeps its smoothing influence even in the infinite dimensional situation. We hope to discuss this in a future paper.     

一类无穷维Hamilton算子的半群生成定理

研究了无穷维H am ilton算子生成C0半群的问题,得到了类无穷维H am ilton算子生成C0半群的一个充分条件.把结果应用在一类双曲型混合问题生成的无穷维H am ilton算子上,证明此类算子生成C0半群,并利用H ille-Y osida定理进一步说明了结果的正确性和有效性.另外,还给出了波动方程相应的无穷维H am ilton算子所生成的C0半群的具体表达式.     

Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

Banach空间中含间断项的二阶非线性常微分方程周期边值问题

在Banach空间中利用上下解方法与不连续增算子不动点定理，研究了含间断项和右端函数具有一阶导数项的二阶非线性常微分方程周期边值问题的最大解、最小解的存在性，推广和改进了现有的结果．而且对于有限维空间，我们获得的这些结果也都是新的．     

Reverse order law for generalized inverses of multiple operator product

In this paper, we consider the reverse order law for generalized inverses of operators on Hilbert spaces. We derive necessary and sufficient conditions for various inclusions concerning the reverse order law for generalized inverses of multiple operator product. We extend the finite dimensional results from (Wei M. Reverse order laws for generalized inverses of multiple matrix products. Linear Algebra Appl. 1999;293:13.) to infinite dimensional settings.     

On the Finite-dimensional Optimal Approximations to the Infinite dimensional Linear Operators

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨａｎｄＫｂｅｒｅａｌｏｒｃｏｍｐｌｅｘｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌＨｉｌｂｅｒｓｐａｃｅ．Ａ（ｂｏｕｎｄｅｄｌｉｎｅａｒ）ｏｐ－ｅｒａｔｏｒＡ：Ｈ→Ｋｉｓｓａｉｄｔｏｂｅｉｎｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ）ｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ）．Ａｉｓｓａｉｄｔｏｂｅｎ－ｄｉｍｅｎｓｉｏｎａｌｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｎ…     

Nilpotent and recursive flows

In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coeficients. Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a generalization of the ‘method of variation of constants’ which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many applications are presented     

Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis.     

只有纯虚谱的Hamilton算子

刻画了一类无穷维Hamilton算子的谱的分布情况,并得到了无穷维Hamilton算子只有纯虚谱的充分条件.最后,构造出具体的例子以说明判别准则的有效性.     

From finitary to infinitary second‐order logic

A back and forth condition on interpretations for those second‐order languages without functional variables whose non‐logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second‐order languages with an infinite non‐logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second‐order languages whose non‐logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second‐order languages can be extended to the infinitary second‐order languages. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.     

Spectral condition numbers for defective elements of linear operators in hilbert spaces

In this paper we generalise matrix eigenelements results on condition numbers to linear bounded operators in complex Hilbert infinite dimensional spaces. Suitable nornls are introduced in order to interpret these results geometrically.     

Semi-linear problems in infinite dimensions

We prove existence, smoothness and ergodicity results for semilinear parabolic problems on infinite dimensional spaces assuming the Logarithmic Sobolev inequality is satisfied. As a consequence we construct a class of nonlinear Markov semigroup which are hypercontractive.     

Dichotomy of Iterated Means for Nonlinear Operators

In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains.     

Existence and Comparison of Green Kernels for Some Elliptic Operators on (0, ∞)

In this paper we give necessary and sufficient conditions for the existence and the comparability of Green kernels for second order elliptic operators in the one dimensional case.     

Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains

Using three different notions of the generalized principal eigenvalue of linear second‐order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.     

On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+[. We show that these operators are the generators of C ₀-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.