DERIVATIONS OF CERTAIN OPERATOR ALGEBRAS

Let be a nest and let be a subalgebra of L(H) containing all rank one operators of alg . We give several conditions under which any derivation δ from into L(H) must be inner. The conditions include (1) H− ≠H, (2) 0+ ≠ 0, (3) there is a nontrivial projection in which is in , and (4) δ is norm continuous. We also give some applications.


Introduction.
In this paper, we unify some results on derivations by considering derivations from an algebra Ꮽ containing all rank one operators of a nest algebra into an Ꮽ-bimodule Ꮾ. Chernoff [1] proves that every derivation from F(H) into L(H) is inner.In [2], Christensen proves that every derivation from a nest algebra into itself or into L(H) is inner.In [3], Christensen and Peligrad show that every derivation of a quasitriangular operator algebra into itself is inner.Knowles [7] generalizes the result of [2] and gets that any derivation from a nest algebra into an ideal of L(H) is inner.Let ᏺ be a nest of subspaces of a Hilbert space H, let Ꮽ be a subalgebra of L(H) containing all rank one operators of alg ᏺ, and let δ be a derivation from Ꮽ into L(H).We prove that if one of the following conditions holds: (1) H − ≠ H, (2) 0 + ≠ 0, (3) there exists a nontrivial P ∈ ᏺ, such that P ∈ Ꮽ, then δ is inner.We also prove that for any nest, if δ is a norm continuous derivation from Ꮽ into L(H), then δ is inner.
We discuss some applications of these results.
Let H be a complex separable Hilbert space, L(H) the algebra of all bounded linear operators on H, K(H) the ideal of all compact operators in L(H), F (H) the subalgebra of all finite rank operators on H, and F 1 (H) the subset of all operators in F(H) with rank less than or equal to 1.We call a subalgebra Ꮽ of L(H) standard provided Ꮽ contains F(H).A collection ᏸ of subspaces of H is said to be a subspace lattice if it contains zero and H, and is complete in the sense that it is closed under the formation of arbitrary closed linear spans and intersections.A subspace lattice ᏺ is called a nest if it is a totally ordered subspace lattice.Given a nest ᏺ, let alg ᏺ = {T ∈ L(H) : T N ⊆ N, N ∈ ᏺ}.Alg ᏺ is said to be the nest algebra associated with ᏺ.If ᏺ is a nest and E ∈ ᏺ, then we define E − = ∨{F ∈ ᏺ : F ⊊ E}, and E + = ∧{F ∈ ᏺ : F ⊋ E}.If e, f ∈ H we write e * ⊗ f for the rank one operator x → (x, e)f , whose norm is e f .If ᏺ is a nest, then by [8,Lemma 3.7], e * ⊗ f ∈ alg ᏺ if and only if there is an E ∈ ᏺ such that f ∈ E and e ∈ (E − ) ⊥ .If Ꮽ is a subalgebra of L(H), then we say that Ꮽ is a triangular operator algebra, if Ꮽ ∩ Ꮽ * is a maximal abelian selfadjoint subalgebra of L(H).If is maximal triangular, and lat Ꮽ is a maximal nest, then we say that Ꮽ is strongly reducible.A derivation δ of an algebra Ꮽ into an Ꮽ-bimodule Ꮾ is a linear map satisfying δ(AB) = Aδ(B)+δ(A)B, for any A, B ∈ Ꮽ.A derivation δ is called Ꮾ-inner if there exists T ∈ Ꮾ, such that δ(A) = AT −T A. When we say that a derivation δ : Ꮽ → Ꮾ is inner, we mean Ꮾ-inner.

Derivations
Let ᏺ be a nest.In the following, we consider the derivation from a subalgebra Ꮽ of L(H) containing all rank one operators of alg ᏺ into L(H). 3) It remains to show that δ is bounded.Let lim n→∞ x n = x, and lim n→∞ T x n = y.Now for any rank one operator A ∈ alg ᏺ, we have that δ(A) and T A are bounded.It follows that AT = δ(A) + T A is bounded, and lim n→∞ AT x n = AT x = Ay.Since Ꮽ contains all rank one operators of alg ᏺ, and by [4,Proposition 3.8], every finite rank operator of alg ᏺ is a sum of some rank one operators of alg ᏺ, we have, for any finite rank operator B in alg ᏺ, BTx = By.By [4, Theorem 3.11], choose a bounded net {B λ } of finite rank operators in alg ᏺ such that lim λ B λ = I in the strong operator topology.We have T x = y.By the closed graph theorem, it follows that T is bounded.
Corollary 2.2.If ᏺ is a nest such that 0 + ≠ 0, and Ꮽ is a subalgebra of L(H) containing all rank one operators of alg ᏺ, then every derivation δ from Ꮽ into L(H) is inner.
It is easy to prove that δ * is a derivation from Ꮽ * into L(H).By Theorem 2.1, we have that δ * is inner.It follows that δ is inner.
We now consider a nest ᏺ such that H such that ST = T S on E 2 for any rank one operator S of alg ᏺ, then there exists a λ such that T x = λx, for any x ∈ E 1 .
Proof.For x ∈ E 1 , choose y ∈ E 2 − E 1 such that y = 1.Since y * ⊗ x ∈ alg ᏺ, by hypothesis (2.4) Since every one-dimensional subspace of L(E 2 ,H) is reflexive, it follows that there exists λ such that T = λI.Proof.Since H − = H, we may choose an increasing sequence {P i } ⊆ ᏺ such that P i → I in the strong operator topology.Also choose f * ∈ P ⊥ i , and y ∈ H, such that f * = 1, f * (y) = 1, and y ≤ 2. Define, ( Using an argument similar to the proof of Theorem 2.1, we may prove that for A in Ꮽ, By Lemma 2.3, we have T j − T i = λ ij on P i−1 .Now for j > 2, let Tj = T 1 + λ 1,j .We have, for k > j > 2, Tj x = Tk x for all x ∈ P j−1 .Now for any x ∈ ∪{P i } = ∪{N : N ⊊ H, N ∈ ᏺ}, choose a j > 2 such that x ∈ P j and let T x = Tj x.Then, T is well defined and for x in M, δ(A)x = (AT − T A)x.
Remark 2.5.Using the idea in the proof of Theorem 2.1, we can prove that in Lemma 2.3, T i is a bounded operator from P i into H.Theorem 2.6.If ᏺ is a nest, Ꮽ is a subalgebra of L(H) containing all rank one operators of alg ᏺ, and δ is a norm continuous derivation from Ꮽ into L(H), then δ is inner.
Proof.If ᏺ satisfies H − ≠ H, then by Theorem 2.1, we get that δ is inner.If ᏺ satisfies H − = H, then by Lemma 2.4, there exists a linear map T such that (2.7) By (2.5) and the boundedness of δ, it follows that T i x ≤ 2 δ x .Since |λ ij | ≤ T i + T j ≤ 4 δ , it follows that T ≤ 6 δ .Thus T is bounded on M. Let T be the unique bounded extension of T to H. Then T is bounded and for A in Ꮽ, δ(A) = A T − T A. Theorem 2.7.Let ᏺ be a nest satisfying H − = H.If there exists a nontrivial projection P ∈ ᏺ, such that P ∈ Ꮽ, and δ is a derivation from Ꮽ into L(H), then δ is inner.
Proof.As in the proof of Lemma 2.4, we choose P 1 = P .Let H = P ⊕ P ⊥ .Then T can be decomposed as By the definition of T , T 11 and T 21 are bounded.We now prove that T 12 and T 22 are bounded.Since A = 1 0 0 0 in Ꮽ, we have that δ(A) = 0 T 12 −T 21 0 holds on M. Since δ(A) is bounded, it follows that T 12 is bounded.Now, for any rank one operator A ∈ L(H), we have P A(1 − P ) ∈ Ꮽ.Hence, holds on M. Since δ(P A(1 − P )) is bounded, it follows that P A(1 − P )T 22 is bounded.
Hence for any f * ∈ P ⊥ and e ∈ P , e ≠ 0, f * ⊗eT 22 is bounded on Q.Thus there exists c such that |f * (T 22 x)| ≤ c, for any x ∈ Q, and x ≤ 1.By the uniform boundedness theorem, we have that { T 22 x : x ≤ 1} is bounded.Hence T 22 is bounded.As in Theorem 2.6, there exists a bounded extension T of T to H such that for A in Ꮽ, δ(A) = A T − T A.

Applications.
In this section, we apply the results above to some special subalgebras of L(H).If A ⊇ F(H), then by Theorem 2.1, we have the following corollaries.Corollary 3.1 [1].Every derivation from a standard operator algebra into L(H) is inner.Corollary 3.2 [2].If δ is a derivation from alg ᏺ into itself, then δ is inner.
Proof.By Theorems 2.1 and 2.7, we have that there is T in L(H) such that for any A in Ꮽ, δ(A) = AT − T A. Now we prove that T is in alg ᏺ.Now for any P in ᏺ, since δ(P ) = P T − T P in alg ᏺ, we have that (I − P )δ(P )P = 0 = −(I − P )T P .This completes the proof.
Using this lemma and Theorem 2.7, we easily prove the following result.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Lemma 2 . 4 .
Let ᏺ be a nest such that H − = H, and let M = ∪{N : N ⊊ H, N ∈ ᏺ}.Then there exists a linear map T from M into H such that δ(A)x = (AT − T A)x, for any x in M.

Corollary 3 . 4 .
If Ꮾ is an algebra containing alg ᏺ, then any derivation δ : Ꮾ → C p is inner for 1 ≤ p ≤ ∞.Corollary 3.5.If Ꮾ is a triangular operator algebra containing every rank one operator in alg ᏺ, then every derivation δ from Ꮾ into L(H) is inner.