Nest Algebras Research Articles

Lie triple and Jordan derivable mappings on nest algebras

Let 𝒩 be a nontrivial nest on a Banach space X over the complex field ℂ, assume that there exists a nontrivial element in 𝒩 which is complemented in X. Let Alg 𝒩 be the associated nest algebra. In this article, we show that if δ is a Lie triple derivable mapping from Alg 𝒩 into B(X) then for any A, B ∈ Alg 𝒩 there exists a λ A,B (depending on A and B) in ℂ such that δ(A + B) = δ(A) + δ(B) + λ A,B I, and δ = D + τ, where D is an additive derivation from Alg 𝒩 into B(X) and τ is a mapping from Alg 𝒩 into ℂI such that τ(A + B) = τ(A) + τ(B) + λ A,B I and τ([[A, B], C]) = 0 for all A, B, C ∈ Alg 𝒩. We also show that if δ is a Jordan derivable mapping from Alg 𝒩 into B(X) then δ is an additive derivation.

N-Derivations of triangular algebras

Let A be a triangular algebra. Let n⩾2 be an integer. A map φ:A×A×⋯×A→A is said to be a n-derivation if it is a derivation in each argument. In this paper we investigate n-derivations (n⩾3) for a certain class of triangular algebras. The main result is then applied to upper triangular matrix algebras and nest algebras.

Characterizations of all-derivable points in nest algebras

Let A \mathcal {A} be an operator algebra on a Hilbert space. We say that an element G ∈ A G\in {\mathcal {A}} is an all-derivable point of A {\mathcal {A}} if every derivable linear mapping φ \varphi at G G (i.e. φ ( S T ) = φ ( S ) T + S φ ( T ) \varphi (ST)=\varphi (S)T+S\varphi (T) for any S , T ∈ a l g N S,T\in alg{\mathcal {N}} with S T = G ST=G ) is a derivation. Suppose that N \mathcal {N} is a nontrivial complete nest on a Hilbert space H H . We show in this paper that G ∈ a l g N G\in {alg\mathcal {N}} is an all-derivable point if and only if G ≠ 0 G\neq 0 .

Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,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 = AΦ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let δ:U →U be an additive map. It is also shown that the following four conditions are equivalent: (1) δ is specially generalized derivable at zero point, i.e., δ(AB) = δ(A)B +Aδ(B)−Aδ(I)B whenever AB = 0; (2) δ 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.

Characterization of higher derivations on CSL algebras

Let AlgL be a CSL algebra. We say that a family of linear maps δ={δn,δn:AlgL→AlgL,n∈N} is higher derivable at Ω∈AlgL if ∑i+j=nδi(A)δj(B)=δn(Ω) for all A,B∈AlgL with AB=Ω. In this paper, a necessary and sufficient condition for a family of linear maps δ={δn,n∈N} on AlgL to be higher derivable at Ω∈AlgL is given. Moreover, we show that if there is a faithful projection P in L such that PΩP and (I−P)Ω(I−P) are a left or right separating point in PAlgLP and (I−P)AlgL(I−P) respectively, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω if and only if it is a higher derivation. In particular, if AlgL is an irreducible CDCSL algebra or a nest algebra, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω≠0 if and only if it is a higher derivation.

Zero product determined some nest algebras

Let AlgN be a nest algebra associated with the nest N on a (real or complex) Hilbert space H. We say that AlgN is zero product determined if for every linear space N and every bilinear map ϕ:AlgN×AlgN→V the following holds: if ϕ(A,B)=0 whenever AB=0, then there exists a linear map T such that ϕ(A,B)=T(AB) for all A,B∈AlgN. If we replace in this definition the ordinary product by the Jordan (resp., Lie) product, then we say that AlgN is zero Jordan (resp., Lie) product determined. We show that any finite nest algebra over a complex Hilbert space is zero product determined, and it is also zero Jordan product determined. Moreover, we show that any finite-dimensional nest algebra on a (real or complex) Hilbert space is zero Lie (resp., associative, Jordan) product determined. In addition, we characterize separately strongly operator topology continuous bilinear map ϕ from AlgN×AlgN into a topological linear space V with the property that ϕ(A,B)=0 whenever AB=0 or ϕ(A,B)=0 whenever AB+BA=0.

Connectedness of the invertibles in certain nest algebras of order type ω

We solve the connectedness problem for a class of nests of order type ω with finite dimensional atoms.

Additive mappings derivable at non-trivial idempotents on Banach algebras

Let 𝒜 be a Banach algebra with unity I containing a non-trivial idempotent P and ℳ be a unital 𝒜-bimodule. Under several conditions on 𝒜, ℳ and P, we show that if d : 𝒜 → ℳ is an additive mapping derivable at P (i.e. d(AB) = Ad(B) + d(A)B for any A, B ∈ 𝒜 with AB = P), then d is a derivation or d(A) = τ(A) + AN for some additive derivation τ : 𝒜 → ℳ and some N ∈ ℳ, and various examples are given which illustrate limitations on extending some of the theory developed. Also, we describe the additive mappings derivable at P on semiprime Banach algebras and C*-algebras. As applications of the above results, we characterize the additive mappings derivable at P on matrix algebras, Banach space nest algebras, standard operator algebras and nest subalgebras of von Neumann algebras. Moreover, we obtain some results about automatic continuity of linear (additive) mappings derivable at P on various Banach algebras.

Applications of Operator Space Theory to Nest Algebra Bimodules

Recently, Blecher and Kashyap have generalized the notion of W*-modules over von Neumann algebras to the setting where the operator algebras are σ closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras. We prove that Y is a right weak* rigged module over a nest algebra \({\rm{Alg}(\mathcal M)}\) if and only if there exists a completely isometric normal representation \({\Phi }\) of Y and a nest algebra \({\rm{Alg}(\mathcal N)}\) such that \({\rm{Alg}(\mathcal N) \Phi (Y)\rm{Alg}(\mathcal M)\subset \Phi (Y)}\) while \({\Phi (Y)}\) is implemented by a continuous nest homomorphism from \({\mathcal M}\) onto \({\mathcal N}\) . We describe some properties which are preserved by continuous CSL homomorphisms.

CHARACTERIZATIONS OF LIE HIGHER AND LIE TRIPLE DERIVATIONS ON TRIANGULAR ALGEBRAS

In this paper, we show that under certain conditions every Lie higher derivation and Lie triple derivation on a triangular algebra are proper, respectively. The main results are then applied to (block) upper triangular matrix algebras and nest algebras.

Derivable maps and derivational points

For an algebra A and an A-bimodule M, let L(A,M) be the set of all linear maps from A to M. A map δ∈L(A,M) is called derivable atC∈A if δ(A)B+Aδ(B)=δ(C), for all A,B∈A with AB=C. We call an element C∈A a derivational point of L(A,M) if ∀δ∈L(A,M) the condition δ is derivable at C implies δ is a derivation. We characterize derivable maps by means of Peirce decompositions and determine derivational points for some general bimodules. As a special case, we see that for a nest algebra A on a Hilbert space H, every 0≠C∈A is a derivational point of L(A,B(H)).

Lie isomorphisms of nest algebras on Banach spaces

We show that a Lie isomorphism ψ between nest algebras on Banach spaces can be decomposed as ψ=ϕ+τ, where ϕ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator.

Multiplicative Lie n-derivations of triangular rings

We introduce the notion of a multiplicative Lie n-derivation of a ring, generalizing the notion of a Lie (triple) derivation. The main goal of the paper is to consider the question of when do all multiplicative Lie n-derivations of a triangular ring T have the so-called standard form. The main result is applied to the classical examples of triangular rings: nest algebras and (block) upper triangular matrix rings.

Lie triple derivations of nest algebras on Banach spaces

Let N be a non-trivial nest on X, AlgN be the associated nest algebra, and L:AlgN→B(X) be a linear mapping. In this paper, it is proved that L is a Lie triple derivation if and only if there exist a derivation d:AlgN→B(X) and a linear mapping h:AlgN→CI with h([[X,Y],Z])=0 for any X,Y,Z∈AlgN such that L=d+h on AlgN.

All-derivable points of nest algebras on Banach spaces

Let N be a nest on a real or complex Banach space X and let AlgN be the associated nest algebra. Ω ∈ AlgN is called an additively all-derivable point if for any additive map δ : AlgN →AlgN , δ (AB) = δ (A)B+Aδ (B) holds for any A,B ∈ AlgN with AB = Ω implies that δ is an additive derivation. Assume that P is an idempotent operator with range ran(P) = N0 for some nontrivial N0 ∈ N . Let Ω ∈ AlgN be any operator satisfying that PΩP =Ω (or (I−P)Ω(I−P) =Ω ). We show that, if Ω|ran(P) (or Ω|ran(I−P) ) is injective or has dense range, then Ω is an additively all-derivable point. Moreover, if X is infinite dimensional, then every additive map derivable at such an Ω is an inner derivation. Mathematics subject classification (2010): 47L35, 47B47.

The Reliable Stabilization Problem for Discrete Time-Varying Linear Systems

In this paper, we consider the reliable stabilization problem for discrete time-varying linear systems using a two-controller configuration in the framework of nest algebras. We establish some sufficient conditions for the existence of solutions of the reliable stabilization problem and obtain the parametrization of the reliable controller pairs for strongly stabilizable linear systems.

Strong commutativity preserving maps on triangular rings

Let U = Tri(A ,M ,B) be a triangular ring. It is shown, under some mild assumption, that every surjective strong commutativity preserving map Φ : U →U (i.e. [Φ(T ),Φ(S)]= [T,S] for all T,S ∈ U ) is of the form Φ(T ) = ZT + f (T ) , where Z is in Z (U ) , the center of U , Z2 = I and f is a map from U into Z (U ) . As an application, a characterization of general surjective maps that preserve the strong commutativity on the nest algebras of Banach space operators is given. Mathematics subject classification (2010): 47L35, 16U80.

Stabilization of Time‐Varying System by Controllers with Internal Loop

We study the concept of stabilization with internal loop for infinite‐dimensional discrete time‐varying systems in the framework of nest algebra. We originally give a parametrization of all stabilizing controllers with internal loop, and it covers the parametrization of canonical or dual canonical controllers with internal loop obtained before. We show that, in practical application, the controller with internal loop overcomes the awkwardness brought by the extra invertibility condition in the parametrization of the conventional controllers. We also prove that the strong stabilization problem can be completely solved in the closed‐loop system with internal loop. Thus the advantage of the controller with internal loop is addressed in the framework of nest algebra.

A decomposition theorem for Lie ideals in nest algebras

Let \({\mathcal {N}}\) be a nest and let \({\mathcal {L}}\) be a weakly closed Lie ideal of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We explicitly construct the greatest weakly closed associative ideal \({\mathcal {J} (\mathcal {L})}\) contained in \({\mathcal {L}}\) and show that \({\mathcal {J} (\mathcal {L}) \subseteq \mathcal {L} \subseteq \mathcal {J} (\mathcal {L})\oplus {\breve{\mathcal{D}}} (\mathcal {L})}\) , where \({{\breve{\mathcal{D}}}} (\mathcal {L})\) is an appropriate subalgebra of the diagonal \({\mathcal {D} (\mathcal {N})}\) of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We show that norm-preserving linear extensions of elements of the dual of \({\mathcal {L}}\) , satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal \({\mathcal {J} (\mathcal {L})}\) .

On stabilization for discrete linear time-varying systems

In this paper, we consider the stabilization problem for discrete linear time-varying systems in an operator-theoretic framework. By using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. This result extends the Youla parametrization theorem by only requiring a strong left or a strong right representation but not both. The strong stabilization problem is also discussed.