Hilbert空间上两个正交射影的乘积

该文研究Hilbert空间H上正则射影对(P,Q)的性质和结构,给出H上有界线性算子A表示为两个正交射影乘积的充分必要条件.     

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature ?1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zero-modes of the two-dimensional Pauli operator.     

Global Hypoellipticity and Simultaneous Approximability

A necessary and sufficient condition is given for a sum of squares operator to be globally hypoelliptic on an N-dimensional torus. This condition is expressed in terms of Diophantine approximation properties of the coefficients. The proof of the Theorem is based on L²-estimates and microlocalization.     

Common Hypercyclic Vectors and the Hypercyclicity Criterion

An operator on a separable, infinite dimensional Banach space satisfies the Hypercyclicity Criterion if and only if the associated left multiplication operator is hypercyclic; see [14], [16], [29]. By examining paths of operators where each operator along the path satisfies the criterion, we provide necessary and sufficient conditions for a path of left multiplication operators to have an SOT-dense set of common hypercyclic vectors. As a corollary, we establish a natural sufficient condition for a path of operators to have a common hypercyclic subspace.     

A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)^–1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.     

Determinimg abelian p-groups from their n-Socles

A necessary condition is given for a very wide separable p-group to be derermined by its n-socle (the set of elements of order at most p ⁿ). A group G is very wide if it has a direct summand of final rank ‖G‖ which itself is a direct sum of cyclic groups. In the case, n=1 (as Shelat has shown)this condition is sufficient but for n>1 examples are given to show that this condition is not sufficient.     

Hilbert空间中具无界控制的无限时区线性二次最优控制问题

本文研究了Hilbert空间中一类由解析半群支配的具无界控制的无限时区线性二次最优控制问题,其中指标中的控制项加权算子要求强制而状态项加权算子可允许为不定号.在指数能稳条件下,证明了任意的最优控制及其最优轨线必定连续,建立了正实引理作为此问题唯一可解的充要条件,并用代数Riccati方程的解给出了最优控制的闭环综合。     

Compact differences of composition operators

A characterization of compact difference is given for composition operators acting on the standard weighted Bergman spaces and necessary conditions are given on a larger scale of weighted Dirichlet spaces. Conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.     

Well-Boundedness of Sums and Products of Operators

A sufficient condition is given under which the sum, productand indeed any polynomial combination of a well-bounded operatorand a commuting real scalar-type spectral operator is well-bounded.This generalizes a result of Gillespie for Hilbert space operators.It is shown in particular that if X is a UMD space, then thesum of finitely many commuting real scalar-type spectral operatorsacting on X is a well-bounded operator (a result which failson general reflexive Banach spaces).     

Set separation, approximating multicones, and the Lipschitz maximum principle

We present a general necessary condition for separation of the reachable set of a Lipschitz control system from another given set of states, expressed in terms of an “approximating multicone” to the set in a sense that contains as special cases the Clarke and Mordukhovich cones. We then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality.     

Banach空间有界线性算子强连续双半群

本文在Ｂａｎａｃｈ空间上研究单参数有界线性算子族－强连续双半群。     

斜对角无穷维Hamilton算子的点谱和特征函数系辛结构的非退化性

本文研究斜对角无穷维Hamilton算子$H=\begin{pmatrix}0&B\\C&0\end{pmatrix}$的点谱和特征函数系辛结构的非退化性, 给出斜对角无穷维Hamilton算子$H$的特征函数系具有非退化辛结构的充分必要条件. 基于此, 进一步刻画了斜对角无穷维Hamilton算子$H$的点谱分别包含于实轴、虚轴以及其它区域的充分必要条件. 最后, 以板弯曲问题和弦振动问题中导出的斜对角无穷维Hamilton算子为例, 验证了所得结论的正确性.     

On the infinite product of operators in Hilbert space

We give a necessary and sufficient condition for a certain set of infinite products of linear operators to be zero. We shall investigate also the case when this set of infinite products converges to a non-zero operator. The main device in these results is a weighted version of the König Lemma for infinite trees in graph theory.

     

Prediction Operators in Banach Ideal Spaces

A necessary and sufficient condition for an operator in a Banach ideal space to be a prediction operator is given.     

上三角无穷维Hamilton算子的剩余谱

将点谱划分为四个部分,得到上三角无穷维Hamilton算子的点谱σ_p(H)关于虚轴对称的充要条件.在此基础上,结合无穷维Hamilton算子的谱结构,得到无穷维Hamilton算子剩余谱的完全描述,从而实现了利用其内部算子刻画剩余谱.     

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…     

Antiderivatives and potentials of local differential operators

A necessary and sufficient condition for the antiderivative of a local differential operator to be local is derived. A formula is given which explicitly exhibits the antiderivative as a local operator when this condition is met. The analogous problem for local vector differential operators is treated. A decomposition of an arbitrary local differential operator in terms of the derivative of a local operator and the gradient of a functional is given.     

上三角无穷维Hamilton算子点谱的对称性

In this paper, by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H, a necessary and sufficient condition is obtained on the symmetry of σP（A） and σ1/P（-A^＊） with respect to the imaginary axis. Then the symmetry of the point spectrum of H is given, and several examples are presented to illustrate the results.     

On existence of cone-maximal points in real topological linear spaces

A sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points. The condition becomes a necessary one in a topological linear space with a countable local base, that is, if the space is pseudometrizable. The paper extends known results to infinite dimensions and we answer Corley’s question in the affirmative with the exception of a pathological case.