MAXIMAL NILPOTENT IDEALS AND ISOMORPHISMS OF LARGE SUBALGEBRAS OF NEST ALGEBRAS

Let N be a nest of projections on a Hilbert space H and T(N) be the corresponding nest algebra. Let A be a large subalgebra of T(N). It is proved that any maximal n-nilpotent ideal of A is in the form of A∩R F,where F is a finite subnest of N and R F is the Jacobson radical of T(F).Using this result can prove that two large subalgebras are isomorphic if and only if the corresponding nests are similar.

Lie Higher Derivations on Nest Algebras

Let N be a nest on a Banach space X, and Alg N be the associated nest algebra. It is shown that if there exists a non-trivial element in N which is complemented in X, then D = （Ln）n∈N is a Lie higher derivation of AlgAl if and only if each Ln has the form Ln（A） ： Tn（A） ＋ hn（A）I for all A ∈ AlgN, where （Tn）n∈N is a higher derivation and （hn）n∈N is a sequence of additive functionals satisfying hn（[A,B]） = 0 for all A,B ∈ AlgN and all n ∈ N.

A Local Characterization of Lie Homomorphisms of Nest Algebras

In this paper,linear maps preserving Lie products at zero points on nest algebras are studied.It is proved that every linear map preserving Lie products at zero points on any finite nest algebra is a Lie homomorphism.As an application,the form of a linear bijection preserving Lie products at zero points between two finite nest algebras is obtained.

Linear Maps Preserving Idempotence on Nest Algebras

In this paper,we discuss the rank-1-preserving linear maps on nest algebras of Hilbert- space operators.We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear bijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism,and in turn,if and only if it is an automorphism or an anti-automorphism.

RANK ONE OPERATORS AND TRIANGULAR ALGEBRAS

In this paper, a necessary condition for a maximal triangular algebra to be closed is given. A necessary and sufficient condition for a maximal triangular algebra to be strongly reducible is obtained.

Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.

ADDITIVE MAPS ON SOME OPERATOR ALGEBRAS BEHAVING LIKE(α, β)-DERIVATIONS OR GENERALIZED(α, β)-DERIVATIONS AT ZERO-PRODUCT ELEMENTS

Let A be a subalgebra of B（X） containing the identity operator I and an idempotent P. Suppose that α,β： A →A are ring epimorphisms and there exists some nest N on 2（ such that α（P）（X） and β（P）（X） are non-trivial elements of N. Let A contain all rank one operators in AlgN and δ ： A→ B（X） be an additive mapping. It is shown that, if δ is （α, β）-derivable at zero point, then there exists an additive （α, β）-derivation τ ： A →β（X） such that δ（A） =τ（A） ＋ α（A）δ（I） for all A∈A. It is also shown that if δ is generalized （α,β）-derivable at zero point, then δ is an additive generalized （α, β）-derivation. Moreover, by use of this result, the additive maps （generalized） （α,β）-derivable at zero point on several nest algebras, are also characterized.

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T（N） acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T（N）-bimodule containing the nest algebra. They also prove that Hcb^n（A, M） ≌ Hcb^n（A, M） for all n ≥ 1 and a CSL algebra .A with an ultraweakly closed .A-bimodul.M containing A.

Cocycle Perturbation on Banach Algebras

Letαbe a flow on a Banach algebra B, and t 7→ut a continuous function from R into the group of invertible elements of B such that usαs（ut）=us＋t, s, t∈R. Then βt =Adut？αt, t∈R is also a flow on B, where Adut（B） , utBu-1t for any B ∈ B. β is said to be a cocycle perturbation of α. We show that if α, β are two flows on a nest algebra （or quasi-triangular algebra）, thenβ is a cocycle perturbation ofα. And the flows on a nest algebra （or quasi-triangular algebra） are all uniformly continuous.

A Note on the Connectedness of the Invertible Group of a Nest Algebra

The connectedness of the invertibles question for arbitrary nest has been reduced to the case of the lower triangular operators with respect to a fixed orthonormal basis en for n 1. For each f ∈ H∞, let Tf be the Toeplitz operator. In this paper we prove that Tf can be connected to the identity through a path in the invertible group of the lower triangular operators if f satisfies certain conditions.

Interpolation Problems for Nest Algebra Modules

Let U be a weakly closed nest algebra module acting on a Hilbert space H. Given two operators X and Y in B(H), a necessary and sufficient condition for the existence of an operator T in U satisfying TX = Y is provided.

Some Remarks on Distributional Chaos for Linear Operators

In this paper, we consider some properties for bounded linear operators concerning distributional chaos. Norm-unimodality of bounded linear operators im- plies distributional chaos. Some properties such as similarity and spectra description for norm-unimodal operators are considered. The existence of distributional chaos in nest algebra is also proved. In addition, we obtain a sufficient and necessary condition of distributional chaos for a class of operators, which contains unilateral backward weighted shift operators.

Commutant Lifting for Optimal Control of Time-varying Systems

This paper uses the commutant lifting theorem for representations of the nest algebra to deal with the optimal control of infinite dimensional linear time- varying systems. We solve the model matching problem and a certain optimal feedback control problem, each of which corresponds with one type of four-block problem. We also obtain a new formula for the optimal performance and prove the existence of an optimal controller.

Characterizing Centralizers and Generalized Derivations on Triangular Algebras by Acting on Zero Product

Let U = Tri（fit, M, B） be a triangular ring, where A and B are unital rings, and M is a faithful （A, B）-bimodule. It is shown that an additive map φ on U is centralized at zero point （i.e., ,φ（A）B = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ（AB） = δ（A）B ＋ Aτ1（B） = τ2（A）B ＋ Aδ（B） whenever AB = 0; （3） δ is a special generalized derivation; （4） δ is a generalized derivation. These results are then applied to nest algebras of Banach space

Derivations and Isomorphisms of Certain Reflexive Operator Algebras

In this paper, we will prove that every derivation of completely distributive subspace lattice (CDS)algebras on Banach space is automatically continuous. This is new even in the Hilbert space case. As an application of this result, we obtain that every additive derivation of nest algebras on Banach spaces is inner. We will also prove that every isomorphism between nest algebras on Banach space is automatically continuous, and in addition, is spatial.

Equivalent Characterization of Centralizers on B(H)

Let H be a Hilbert space with dimH ≥2 and Z ∈ B（H） be an arbitrary but fixed operator. In this paper we show that an additive map （I） ： B（H）→ B（H） satisfies Ф（AB） = Ф（A）B = AФ（B） for any A, B ∈ B（H） with AB = Z if and only if Ф（AB） = Ф（A）B = AФ（B）, A, B ∈ B（H）, that is, （I） is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Ф（A2） = AФ（A） = Ф（A）A for any A ∈ B（H） with A2 = 0 if and only if Ф（A） = AФ（I） = Ф（I）A, A ∈ B（H）, and generalize main results in Linear Algebra and its Application, 450, 243-249 （2014） to infinite dimensional case. New equivalent characterization of centralizers on B（H） is obtained.

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B（H） denote the Banach algebra of all bounded linear operators on H. For A, B C B（H）, define the binary relation A ≤ B by A＊A = A＊B and AA＊ = AB＊. Then （B（H）, ＂〈.＂） is a partially ordered set and the relation ＂≤＂ is called the star order on B（H）. Denote by Bs（H） the set of all self-adjoint operators in B（H）. In this paper, we first characterize nonlinear continuous bijective maps on Bs（H） which preserve the star order in both directions. We characterize also additive maps （or linear maps） on B（H） （or nest algebras） which are multiplicative at some invertible operator.

Simultaneous Stabilization for a Family of Plants

In this paper, we discuss the problem of simultaneous stabilization for plants more than three by using Youla parametrization and give a necessary and sufficient condition for simultaneous stabilization.

On the Optimal Controller for LTV Measurement Feedback Control Problem

In this paper,we consider the measurement feedback control problem for discrete linear time-varying systems within the framework of nest algebra consisting of causal and bounded linear operators.Based on the inner-outer factorization of operators,we reduce the control problem to a distance from a certain operator to a special subspace of a nest algebra and show the existence of the optimal LTV controller in two different ways：one via the characteristic of the subspace in question directly,the other via the duality theory.The latter also gives a new formula for computing the optimal cost.

On Stabilization for Linear Time-Varying Systems

This paper deals with the stabilization problem for linear time-varying systems within the framework of nest algebras. We give a necessary and sufficient condition for a class of plants to be stabilizable, and we also study the simultaneous and strong stabilization problems.