渐近非扩张非自映射的收敛定理(英文)

设E是实的一致凸Banach空间,K是E的一个非空闭凸集,P是E到K上的非扩张的保核收缩映射.设T1,T2,T3:K→E分别是具有数列{hn},{ln},{kn}[1,∞)的渐近非扩张非自映射,使得sum (hn-1) from n=1 to ∞<∞,sum ((ln-1)) from n=1 to ∞<∞及sum (n=1(kn-1) from n=1 to ∞<∞,且F=F(T1)∩F(T2)∩F(T3)={x∈K:T1x=T2x=T3x}≠Ф.定义迭代序列{xn}:x1∈K,xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1zn),zn=P((1-γn)xn+γnT3(PT3)n-1xn),其中{αn},{βn},{γn}[ε,1-ε],ε是大于零的实数.(i)如果T1,T2,T3中有一个是全连续的或者半紧的,则{xn}强收敛于某一点q∈F;(ii)如果E具有Frechet可微范数或者满足Opial’s条件或者E的对偶空间E~*具有Kadec-Klee性质,则{xn}弱收敛于某一点q∈F.     

区间上三段单调扩张自映射的周期轨道

设ｎ３是奇数，ｍ０是整数，Ｓｎ及ｒｎ分别是方程ｘｎ－２ｘｎ－１－１＝０及ｘｎ＋２－３ｘｎ－ｘ２－１＝０的唯一正根，记ｔｎ０＝ｒｎ，ｔｎｉ＝ｓｎ（ｉ１）．又设ｆ及ｇ分别是闭区间Ｉ上的Ｎ型（即增－减－增型）及反Ｎ型（即减－增－减型）扩张自映射．本文证明了，若ｆ（或ｇ）的扩张常数或则ｆ（或ｇ）有２ｍ·ｎ周期点．此外，本文还指出，当或时，在［０，１］上存在着具有扩张常数λ却无２ｍ·ｎ周期轨道的Ｎ型（或反Ｎ型）扩张自映射．     

向量值函数空间中J-对称算子的J-自伴延拓

给出了向量值函数空间中J-对称算子的J-自伴延拓的完全解析描述.我们应用Knowles理论,借助方程τ(y)=λ     

具有正则型点的奇异微分算子的自共轭扩张

在П(L0)n R≠θ的条件下,本文讨论了具有中间亏指数的对称微分算式l(y)的自共轭域,其中П(L0)是由l(y)生成的最小算子L0的正则型域.使用方程l(y)=λ0y,(λ0∈П(L0)∩R)的实参数L2-解,我们对最大算子域DM进行新的分解,由此得到l(y)的自共轭域新的完全解析刻画,其中自共轭边界条件中矩阵M,N的确定与l(y)=λ0y在无穷远点的性质无关,仅与其在t=0点初始值的选择有关.由于自共轭箅子谱是实的,使用实参数λ0不仅有利于我们找到方程的显解,更重要的是可以得到谱的有关信息.     

共轭拟Fourier-Jacobi级数(英文)

讨论了共轭拟Fourier-Jacobi级数.对于α=β，给出精确的结果。     

区间上满的扩张Markov自映射的迭代根

考虑区间上满的扩张 Markov自映射 ,给出了区间上满的扩张 Markov自映射具有指定阶的迭代根的充分必要条件 .     

区间上平顶单峰扩张自映射的周期轨道

设ｔ（０＜ｔ＜１）是一个常数，ｎ≥３是奇数，ｍ≥０及ｋ≥１是整数，Ｐ０（ｘ）＝ｘ－１，Ｐｉ（ｘ）＝（ｘ２ｉ－１－１）Ｐｉ－１（ｘ）（ｉ≥１），ｒｍｎ（ｔ）及ｒｋ（ｔ）分别是方程Ｐｍ（ｘ）（ｘ２ｍｎ－２ｘ２ｍ（ｎ－２）－１）－ｔ（ｘ２ｍｎ－１）（ｘ２ｍ＋１）＝０及Ｐｋ－１（ｘ）－ｔ（ｘ２ｋ－１＋１）＝０在（１，＋∞）上的唯一实根，ｆ是闭区间Ｉ＝［０，１］上的峰顶区间长度为ｔ的平顶单峰扩张自映射．本文证明了，若ｆ的扩张常数λ≥ｒｍｎ（ｔ）（或＞ｒｋ（ｔ）），，则ｆ有２ｍｎ（或２ｋ）周期点．此外，本文还指出，当１＜λ＜ｒｍｎ（ｔ）（或≤ｒｋ（ｔ）时，在Ｉ上存在着具有扩张常数λ及峰顶区间长度ｔ却无２ｍｎ（或２ｋ）周期点的平顶单峰扩张自映射     

区间上单峰扩张自映射周期轨道的超旋转对

设f是区间I=[0,1]上的单峰扩张自映射, k ∈N,m≥2,λm,k是方程x(k-1)m(xm- 1)Q(x,m 1) (x(k-1)m-1)Q(x,m)=0在(1, ∞)上的唯一实根,其中Q(x,m)=(xm- 2xm-1 1).本文证明:若f的扩张常数λ≥λm,k,则f有超旋转对为(k,km 1)的周期轨道. 此外,还指出,当1<λ<λm,k时,在区间上存在单峰扩张自映射具有扩张常数λ却无超旋转对为(k,km 1)的周期轨道.     

区间上单峰扩张自映射周期轨道的超旋转对

设f是区间I=[0,1]上的单峰扩张自映射,k∈N,m≥2,λm,k是方程x(k-1)m(xm-1)Q(x,m+1)+(x(k-1)m-1)Q(x,m)=0在(1,+∞)上的唯一实根,其中Q(x,m)=(xm-2xm-1+1).本文证明若f的扩张常数λ≥λm,k,则f有超旋转对为(k,km+1)的周期轨道.此外,还指出,当1＜λ＜λm,k时,在区间上存在单峰扩张自映射具有扩张常数λ却无超旋转对为(k,km+1)的周期轨道.     

奇异线性哈密顿算子自伴扩张的新描述

研究了具有一个奇异端点的线性哈密顿算子的白伴扩张的解析描述.设最小哈密顿算子h的亏指数为(d,d),将Im(h~*y,y)表示为秩为2d的二次型,该文利用二次型的表示矩阵得到了最小哈密顿算子h的自伴扩张域的一种新的完全描述.     

奇型Sturm-Liouville微分算子的限界自伴扩张

本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件.     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

Restrictions of Self-Adjoint Partial Isometries

Periodica Mathematica Hungarica -     

二阶微分算子积的自伴性

本文讨论了由正则和奇异的二阶对称微分算式生成的微分算子的积算子的自伴性,得到了积算子为自伴算子时边条件应满足的充分而必要的条件及若干其他结果．     

Closure Operators and Lattice Extensions

For closure operators Γ and Δ on the same set X, we say that Δ is a weak (resp. strong) extension of Γ if Cl(X, Γ) is a complete meet-subsemilattice (resp. complete sublattice) of Cl(X, Δ). This context is used to describe the extensions of a finite lattice that preserve various properties. This revised version was published online in September 2006 with corrections to the Cover Date.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

对称常微分算子的两种自伴延拓形式之间的联系

建立了对称常微分算子von N eum ann自伴延拓与以边条件形式表达的自伴延拓之间的对应关系,给出了相互转换的方法.