Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.     

Invariant Subspaces for Semigroups of Algebraic Operators

T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable.     

Triangularization of a Jordan algebra of Schatten operators

We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.

     

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.     

一类基于量子程序理论的序列效应代数

空间上的算子理论是量子力学的基本数学框架之一.Hilbert空间效应代数是指小于等于单位算子的正算子集合.我们引入了Hilbert空间效应代数的一类子序列效应代数，并讨论了其上序列积的基本运算性质.我们发现：由于代数结构的不同，这类新的序列效应代数与现有效应代数上的运算性质有很大差异.     

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.     

Products of square-zero operators

We characterize matrices that can be written as a product of two or three square-zero matrices. We also consider the same questions for (bounded) operators on an infinite-dimensional, separable, complex Hilbert space and in the Calkin algebra.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?     

Lie algebras with an algebraic adjoint representation revisited

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.     

Multilinear localization operators

In this paper we introduce a notion of multilinear localization operators. By reinterpreting these operators as multilinear Kohn-Nirenberg pseudodifferential operators, we give sufficient conditions for their boundedness on products of modulation spaces. Moreover, we prove that these conditions are also necessary for the boundedness of the operators. Finally, as application, we construct various examples of bounded multilinear pseudodifferential operators.     

Isolated points and essential components of composition operators on

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

     

B(X)上保持约当积的固定点的满射

设B(X)是维数大于等于3的复Banach空间X上有界线性算子全体构成的代数.设A∈B(X),若Ax=x,则称x∈X是算子A的固定点.Fix(A)表示A的所有固定点的集合.本文刻画了B(X)上保持算子的Jordan积的固定点的满射.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Linearly compact algebraic Lie algebras and coalgebraic Lie coalgebras

It is proved that if the dual Lie algebra of a Lie coalgebra is algebraic, then it is algebraic of bounded degree. This result is an analog of the D.Radford's result for associative coalgebras.

     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

保持算子乘积投影的线性映射

设B(H)是复Hilbert空间H上的有界线性算子全体且dim H≥2.本文证明了B(H)上的线性满射φ保持两个算子乘积非零投影性的充分必要条件是存在B(H)中的酉算子U以及复常数λ满足λ~2=1,使得φ(X)=λU~*XU,(?)X∈B(H).同时也得到了线性映射保持两个算子Jordan三乘积非零投影的充分必要条件.     

Weyl代数研究简介

本文简要综述Ｗｅｙｌ代数诞生７０余年来的一系列重要研究成果。     

On the bounded compact approximation property and measures of noncompactness

The Banach spaces with the bounded compact approximation property are characterized by the completeness of measures of noncompactness. Applications to the theory of semi-Fredholm operators and to the algebraic properties of their canonical images in the Calkin algebra are discussed.     

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

Abstract

The notation of I–convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I–convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces.     

Larson理想中算子的插值

设a为nest代数，R^∞为A的Larson理想，R(A)为A的根，[R(A)]^s为R（A）的强拓扑闭。在文本中，我们给出[R(A)]^s的纯代数构造；且引进了一个新算子集合I，并证明了：若A的不变子空间格为几乎原子的，则R^∞＝[R(A)]^s=I。利用上述结果，我们研究了当A的不变子空间格为几乎原子时的Larson理想中的算子插值问题。我们得到算子方程AX＝Y在R^∞中有解A的充分必要条件。