算子的拟酉等价性

本文在复的，可析的、无限维Ｈｉｂｅｒｔ空间Ｈ中引进了算子拟酉等价性的概念；获得了几个重要结果。     

算子多项式的若干性质

本文讨论了多元算子多项式的几种保正号的性质，并由此得出关于正规算子不等式的转化定理。     

关于某些次正规算子的表示

设T是完全非正规的次正规算子。给出了T＝N＋K的充要条件，其中N是正规算子，K是拟正规算子，且NK＝KN。     

有界线性算子乘积以及和的广义Drazin可逆性

本文讨论了两个有界线性算子的乘积以及和的广义Drazin可逆性及其广义Drazin逆的表达式.在新条件下,采用空间分解的方法证明了算子乘积PQ以及算子和P+Q是广义Drazin可逆的,并给出(PQ)^d和(P+Q)^d的具体表达式.     

图象的自反与渐近自反

本文对B（X,Y）的自反子空间及渐近自反子空间上的映射（或映射组）,分别给出了一些判别它们的图象（或联合图象）仍为自反及渐近自反的方法。     

映射Fp：X→‖AXB—C‖p^p的临界点的特征

对于Ｈｉｌｂｅｒｔ空间有界线性算子Ａ、Ｂ、Ｃ,考虑了当Ａ有一个广义逆Ａ＾－使得（ＡＡ＾－）＾＊＝ＡＡ＾－,Ｂ有一个广义逆Ｂ＾－Ｔ使得（Ｂ＾－Ｂ）＾＊＝Ｂ＾－Ｂ时,映射Ｆｐ：Ｘ→‖ＡＸＢ－Ｃ‖ｐ＾ｐ临界点的特征的一般形式（１〈ｐ〈∞）,推广了Ｐ．Ｊ．Ｍａｈｅｒ的关于ｐ＝２时的结果,并指出该定理可推广到多个算子的情形。     

连续小波变换及小波框架算子的一些性质

以泛函分析的观点来考察连续小波变换及小波框架算子，得到了它们的一些性质，并给出了严格证明，弥补了有关献中的不足。     

Decomposition of Analysis Operators and Frame Ranges for Continuous Frames

In this article, the measure space associated with a continuous frame is supposed to be σ-finite and positive, and a frame range is the range of the analysis operator for a continuous frame. Gabardo and Han in 2003 asked whether two frame ranges can both be contained in another one. To solve this problem, we give two decompositions of analysis operators and frame ranges for continuous frames respectively, which essentially establish a relationship between continuous frames and Hilbert-Schmidt operator valued frames. As applications, it follows that only separable Hilbert space can have a continuous frame, that there exists a continuous frame of Riesz-type if and only if the associated measure space is purely atomic, and that the sum of two frame ranges is still a frame range when the sum is closed. Finally, we construct a counterexample which shows that the Gabardo-Han problem is not necessarily true in general.     

广义框架和框架算子

本文研究了可分的Hilbert空间H中的广义框架,应用算子论方法给出了广义框架是H中紧广义框架,对偶广义框架,独立广义框架的充要条件:证明了有关广义框架算子的一些结果。     

Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element.     

幂等算子线性组合的幂等性

设P与Q是Hilbert空间中的两个不同的幂等算子.本文主要刻画了幂等算子P与Q的线性组合仍是幂等算子的充要条件,从而推广了Baksalary与Baksalary (2000)的结论.值得指出的是,我们通过严密的推理发现,其定理的条件P_1P_2≠P_2P_1是非必要的.     

Analysis of Algorithms for Orthogonalizing Products of Unitary Matrices

We consider the problem of computing U_k = Q_kU_k−1(where U₀ is given) in finite precision (ϵ_M = machine precision) where U₀ and theQ_i are known to be unitary. The problem is that Û_k, the computed product may not be unitary, so one applies an O(n²) orthogonalizing step after each multiplication to (a) prevent Û_k from drifing too far from the set of untary matrices (b) prevent Û_k from drifting too far from U_k the true product. Our main results are 1. Scaling the rows to have unit length after each multiplication (the cheaptest of the algorithms considered) is usually as good as any other method with respect to either of the criteria (a) or (b). 2. A new orthogonalization algorithm that guarantees the distance of Û_k (k = 1, 2, …) to the set of unitary matrices is bounded by n^3.5ϵ_M for any choice of Q_i.     

Nonlinear Commutation Relations: Representations by Point-Supported Operators

We present a class of non-Lie commutation relations admitting representations by point-supported operators (i.e., by operators whose integral kernels are generalized point-supported functions). For such relations we construct all operator-irreducible representations (up to equivalence). Each representation is realized by point-supported operators in the Hilbert space of antiholomorphic functions. We show that the reproducing kernels of these spaces can be represented via hypergeometric series and the theta function, as well as via their modifications. We construct coherent states that intertwine abstract representations with irreducible representations.     

Simultaneously Definitizable and Spectralizable Operators in Krein Space

We study the spectral properties of a self–adjoint operator in Krein space such that one of its natural powers is J–non–negative and some other power is a spectral operator in the Dunford sense.     

Unitary Representations of the q-Virasoro Algebras

In the present paper we define an antilinear anti-involution ωof the q-Virasoro (Virq for short), and an Hermitian contravariant form of a Virq-module. For the irreducible module Vq(a,b) over Virq we give a necessary and sufficient condition for Vq(a,b) having a non-degenerate Hermitian contravariant form, and a necessary and sufficient condition for the Hermitian form being unitary.     

A Rank Theorem of Operators between Banach Spaces

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T ₀∈B(E, F) with a generalized inverse T ₀ ⁺∈B(F, E). This paper shows that, for every T∈B(E, F) with ‖T ₀ ⁺ (T−T ₀)‖<1, B ≡ (I + T ₀ ⁺(T−T ₀))⁻¹ T ₀ ⁺ is a generalized inverse of T if and only if (I−T ₀ ⁺ T ₀)N(T) = N(T ₀), where N(·) stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T ₀ is a semi-Fredholm operator but not in general.