On Some Functional Equations on Standard Operator Algebras
نویسندگان
چکیده
The main purpose of this paper is to prove the following result. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, let A(X) ⊆ L(X) be a standard operator algebra, and let T : A(X) → L(X) be an additive mapping satisfying the relation T (A2n+1) = 2n+1 ∑ i=1 (−1)i+1Ai−1T (A)A2n+1−i, for all A ∈ A(X) and some fixed integer n ≥ 1. In this case T is of the form T (A) = AB+BA, for all A ∈ A(X) and some fixed B ∈ L(X). In particular, T is continuous. Throughout, R will represent an associative ring. Given an integer n > 1, a ring R is said to be n-torsion free, if for x ∈ R, nx = 0 implies x = 0. An additive mapping x 7→ x∗ on a ring R is called an involution if (xy)∗ = y∗x∗ and x∗∗ = x hold for all pairs x, y ∈ R. A ring equipped with an involution is called a ring with involution or ∗-ring. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ B. In case we have a ring R an additive mapping D : R → R is called a derivation if D(xy) = D(x)y+xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = ax− xa holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [7] asserts that any 2000 Mathematics Subject Classification. 46K15, 39B05.
منابع مشابه
Lie-type higher derivations on operator algebras
Motivated by the intensive and powerful works concerning additive mappings of operator algebras, we mainly study Lie-type higher derivations on operator algebras in the current work. It is shown that every Lie (triple-)higher derivation on some classical operator algebras is of standard form. The definition of Lie $n$-higher derivations on operator algebras and related pot...
متن کاملOn the superstability of a special derivation
The aim of this paper is to show that under some mild conditions a functional equation of multiplicative $(alpha,beta)$-derivation is superstable on standard operator algebras. Furthermore, we prove that this generalized derivation can be a continuous and an inner $(alpha,beta)$-derivation.
متن کاملSome Properties of $ ast $-frames in Hilbert Modules Over Pro-C*-algebras
In this paper, by using the sequence of adjointable operators from pro-C*-algebra $ mathcal{A} $ into a Hilbert $ mathcal{A} $-module $ E $. We introduce frames with bounds in pro-C*-algebra $ mathcal{A} $. New frames in Hilbert modules over pro-C*-algebras are called standard $ ast $-frames of multipliers. Meanwhile, we study several useful properties of standard $ ast $-frames in Hilbert modu...
متن کاملOn a functional equation for symmetric linear operators on $C^{*}$ algebras
Let $A$ be a $C^{*}$ algebra, $T: Arightarrow A$ be a linear map which satisfies the functional equation $T(x)T(y)=T^{2}(xy),;;T(x^{*})=T(x)^{*} $. We prove that under each of the following conditions, $T$ must be the trivial map $T(x)=lambda x$ for some $lambda in mathbb{R}$: i) $A$ is a simple $C^{*}$-algebra. ii) $A$ is unital with trivial center and has a faithful trace such ...
متن کاملQuasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions
We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...
متن کامل