容许次于最大维不变子空间的三阶非线性微分算子的分类（英文）

利用不变子空间方法研究一般的三阶非线性微分算子的分类问题.证明了当三阶算子容许次于最大维(六维)不变子空间时,它可以被表示为各参量的平方形式,得到了常系数三阶非线性微分算子在六维子空间的完全分类.最后通过一些例子演示利用不变子空间方法约化方程及求精确解的过程.     

序列次可分解算子的不变子空间格

本文研究了序列次可分解算子的不变子空间问题,得到了一类序列次可分解算子具有丰富的不变子空间格的结果,精细地刻划了这类序列次可分解算子的不变子空间格.     

Krein空间中无穷维Hamilton算子的极大确定不变子空间的存在性问题

本文研究了不定度规空间空间中的无穷维Hamilton算子.利用Plus算子存在极大不变子空间的性质,获得了无穷维Hamilton算子在Krein空间中存在极大确定不变子空间的充分条件.     

某些加权复合算子之不变子空间的存在性

本文研究某些加权复合算子之非平凡不变子空间的存在性.特别地,证明了每个亚正规加权复合算子均有非平凡的不变子空间并且提出了一个新概念,称其为本性可逆变换.对于概率空间上本性可逆变换所确定的加权复合算子,给出其非平凡不变子空间存在性的一个等价刻画.     

关于序列次可分解算子的不变子空间

得到了关于序列次可分解算子的一个不变子空间定理，推广了H.Mohebi和M.Rajiabalipour在1994年得到的一个不变子空间定理，并且举例说明存在l2上的有界线性算子T。它有无穷多个变子空间，但是它的不变子空间格Lat(T)不丰富。     

序列次可分解算子的不变子空间格

本文研究了序列次可分解算子的不变子空间问题，得到了一类序列次可分解算子具有丰富的不变子空间格的结果，精细地刻划了这类序列次可分解算子的不变子空间格．     

Sobolev圆盘代数的不变子空间

研究了Sobolev圆盘代数R(D)上乘自变量算子M_z的不变子空间,给出了M_z在任何不变子空间上限制的基本性质,证明了M_z分别限制在两个不变子空间上酉等价当且仅当这两个不变子空间相等,并描述了M_z的一类公共零点在边界的不变子空间的结构.     

多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.     

J-次正常算子的不变子空间

本文证明n阶J-次正常算子在一个亏维数不超过n的不变子空间的限制是次正常算子,且当n>O时这类算子有非平凡超不变子空间。由此易知J-次正常算子有非平凡不变子空间。我们还讨论了拟幂零的和紧的J-次正常算子。     

次可分解算子的不变子空间格的丰富性

证明了一类次可分解算子的不变子空间格是丰富的,并举例说明存在Hilbert空间上的有界线性算子T,它有无穷多个不变子空间,但是它的不变子空间格Lat(T)不丰富．     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Backward errors of periodic invariant subspaces for regular periodic matrix pairs

In this paper, the backward error of periodic invariant subspaces for regular periodic pairs is defined and its explicit expression is derived. In particular, we also present the expression of the backward error of generalized invariant subspaces for the regular matrix pair. The results are illustrated by two numerical examples.     

Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.     

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations.The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit.A two-component nonlinear system of dissipative equations is analyzed to shed light on the resulting theory,and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.     

Regular Factorization of the Scalar Characteristic Function

It is known that regular factorizations of the characteristic function of an operator describe its invariant subspaces. The case of a scalar characteristic function is considered. Some examples are given. The factorizations describing all chains of invariant subspaces containing a given subspace L are constructed by the factorization describing L. A representation of the regular factorization of a function is obtained in terms of factorizations of its inner and outer parts. Bibliography: 9 titles.     

一般非齐次非线性扩散方程的等价变换和高维不变子空间

结合压力变换和不变子空间方法中的等价变换,给出了一般非齐次非线性扩散方程的等价方程,并给出了等价方程的高维不变子空间.由此构造了一般非齐次非线性扩散方程的广义分离变量解,并给出了几个例子解释这个过程.     

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \mathbb T{{\mathbb T}}. Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.     

Controlled and conditioned invariant subspaces in linear system theory

The concept of invariance of a subspace under a linear transformation is strongly connected with controllability and observability of linear dynamical systems. In this paper, we definecontrolled andconditioned invariant subspaces as a generalization of the simple invariants, for the purpose of investigating some further structural properties of linear systems. Moreover, we prove some fundamental theorems on which the computation of the above-mentioned subspaces is based. Then, we give two examples of practical application of the previous concepts concerning the determination of the constant output and perfect output controllability subspaces.     

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

We consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells.