Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

Holomorphic Banach vector bundles on the maximal ideal space of H∞ and the operator corona problem of Sz.-Nagy

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space $M\left(H^{\infty }\right)$ of the Banach algebra $H^{\infty }$ of bounded holomorphic functions on the unit disc $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem.     

On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the $C_0$-semigroup $S\left(t\right)$ generated by $A$ is surjective and the pair $(A,B)$ with a bounded operator $B$ is completely stabilizable, then $(A,B)$ is exactly controllable without any additional condition.     

Serial factorizations of right ideals

In a Dedekind domain D, every non-zero proper ideal A factors as a product $A=P_1^{t_1}?P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain D, the D-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal A of an arbitrary ring R as a product of proper right ideals $A_1,\dots ,A_n$ with all the modules $R/A_i$ uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of h-local Prüfer domains and that of semirigid commutative GCD domains.     

Un contre-exemple pour un espace d'interpolation qui n'est pas faiblement LUR

According to a previous result of the author, if $(A_0,A_1)$ is an interpolation couple, if $A_0^{?}$ is weakly LUR, then the complex interpolation spaces ${(A_0^{?},A_1^{?})}_{\theta }$ have the same property.Here we construct an interpolation couple $(B_0,B_1)$ where $B_0$ is LUR, but where the complex interpolation spaces ${(B_0,B_1)}_{\theta }$ are not strictly convex.