Extensions of normal operators

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.Partially supported by NSF Grant DMS-9107137     

Reflexivity of the commutant and local commutants of an algebraic operator

We show that an algebraic operator on a complex Banach space has reflexive commutant if and only if each zero of the minimal polynomial of the operator is simple. Further, for any operator, the local commutant at an eigenvector is reflexive. On the other hand, for an algebraic operator whose minimal polynomial has at least one zero that is not simple, the local commutant of the operator at a given vector is reflexive precisely when the vector is an eigenvector.     

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.This work was partially supported by NSF Grant DMS-9401544.Participant, Workshop in Linear Analysis and Probability, Texas A&M University     

具有分解性质的次对角代数

设M是σ-有限的von Neumann代数,21是M的具有分解性质的次对角代数,即对任意可逆算子T∈M,都存在西算子U∈M及可逆算子A∈21∩21~(-1),使得T=UA,本文证明了21的代数换位是自伴的,同时也证明了21中的可逆算子群是σ-弱连通集。     

一类算子矩阵特征值的代数指标和代数重数

自伴算子特征值的几何重数与代数重数相等,但对于非自伴算子不一定成立,这主要是特征值的代数指标起着决定性的作用.讨论了一类非自伴算子矩阵特征值的几何重数,代数指标与代数重数.     

On cyclicity of commutants

In this note an example is given of a Hilbert space operator whose commutant is not finitely cyclic. The example settles in the negative a conjecture of D. A. Herrero.     

一类四阶Hamilton算子特征值的代数指标

研究了一类四阶Hamilton算子H_A特征值的代数指标问题.根据算子A与Hamilton算子H_A的关系,讨论了Hamilton算子H_A特征值的几何重数,代数指标及代数重数.最后结合例子说明其结果的有效性.     

Operator algebras with strictly cyclic vectors

We obtain results concerning the invariant subspaces of strictly cyclic operator algebras. In particular, we show that transitivity of a strictly cyclic algebra implies its strict (and hence even its strong) density.Translated from Matematicheskie Zametki, Vol. 16, No. 2, pp. 253–257, August, 1974.     

K 0-group and similarity of operator weighted shifts

By using simultaneous triangularization technique, similarity for operator weighted shifts with finite multiplicity is characterized in terms of K0-group of commutant algebra. The result supports the conjecture posed by Cao et al. in 1999. Moreover, we discuss the relations between similarity and quasi-similarity for operator weighted shifts.     

Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction

We give an operator algebraic formulation of the stabilizer formalism for error correction in quantum computing. The approach relies on an analysis of commutant structures, and gives a natural extension of the classic stabilizer formalism to the general case of arbitrary (not necessarily abelian) Pauli subgroups and subsystem codes. We show how to identify the largest stabilizer subsystem for every Pauli subgroup and discuss examples.     

严格循环与强不可约分解

Let W be an injective unilateral weighted shift, and let W(n) be the orthogonal direct sum of n copies of W. In this paper, we prove that, if the commutant of W is strictly cyclic, then W(n) has a unique (SI) decomposition with respect to similarity for every natural number n.     

A lifting theorem for unbounded quasinormal operators

Relationships between minimal normal extensions of spectral and cyclic type of an unbounded quasinormal operator are discussed and some properties such as, for example, tightness of such extensions are established. A Yoshino type criterion on the lifting of the strong commutant of an unbounded quasinormal operator is proved.     

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.Dedicated to Israel Gohberg, as a token of admiration for his inspiring work in analysis and operator theory, with its far reaching applications, in friendship and with great affection.     

Central intertwining lifting,maximum entropy and their permanence

The central intertwining lifting is used to establish a maximum principle for the commutant lifting theorem. This maximum principle is used to prove that the central intertwining lifting is also a maximal entropy solution for the commutant lifting theorem, when T is a unilateral shift of finite multiplicity. The maximum principle is based on the residual spaces for intertwining liftings, and is motivated by Robinson's minimum energy delay principle for outer functions. A permanence property for the central intertwining lifting is also given.     

Property (R) under Perturbations

Property (R) holds for a bounded linear operator ${T \in L(X)}$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI ? T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.     

The algebraic multiplicity of the complex eigenvalues of population operator and its application

In this paper we discuss the algebraic multiplicity of the complex eigenvalue of population operator. Under certain condition we first prove that all the complex eigenvalues of this operator, except at most finitely many ones, are of algebraic multiplicity 1,and then, as an application of this result, we obtain the asymptotic expansion of the solution of corresponding population system.     

Lyusternik-Schnirelman theory and eigenvalue problems for monotone potential operators

We treat the eigenvalue problem Ax = λBx, where A and B are odd potential operators, A is strictly monotone, bounded, coercive, and continuously invertible, and B is monotone and compact. A naturally defined iteration operator is employed, together with the Lyusternik-Schnirelman theory, to prove the existence of infinitely many nontrivial eigenfunctions. With the possible exception of the multiplicity assertion the results which we obtain are not new. The method which we use, however, has not been applied before to problems of this type. It exploits both the potential character and the monotonicity of the operators and makes the treatment of the infinite dimensional problem essentially as simple as that of its finite dimensional analog. This simplification results primarily from the compactness properties of the iteration operator.     

弹性理论中上三角无穷维Hamilton算子根向量组的完备性

考虑弹性力学中一类上三角无穷维 Hamilton 算子．首先,给出此类Hamilton算子特征值的几何重数和代数指标,进而得到代数重数．其次,根据Hamilton算子特征值的代数重数确定其特征(根)向量组完备的形式,得到此类Hamilton算子特征(根)向量组的完备性是由内部算子特征向量组决定．最后,将所得结果应用到弹性力学问题中．     

上三角无穷维Hamilton算子特征值的代数指标

无穷维Hamilton算子特征函数系是否完备与其代数指标有关,研究了上三角无穷维Hamilton算子特征值的代数指标问题,基于主对角元的特征值和特征向量的某些性质,得到上三角无穷维Hamilton算子的几何重数和代数重数.