A remark concerning commutativity modulo radical in Banach algebras

Pavla Vrbová

Displaying similar documents to “A remark concerning commutativity modulo radical in Banach algebras”

Spectral characterizations of central elements in Banach algebras

Matej Brešar, Peter Šemrl (1996)

Studia Mathematica

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Let A be a complex unital Banach algebra. We characterize elements belonging to Γ(A), the set of elements central modulo the radical. Our result extends and unifies several known characterizations of elements in Γ(A).

Commutators of quasinilpotents and invariant subspaces

A. Katavolos, C. Stamatopoulos (1998)

Studia Mathematica

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It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element...

Concerning spectral characterizations of the radical in Banach algebras

Jaroslav Zemánek (1976)

Commentationes Mathematicae Universitatis Carolinae

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Polynomially compact derivations on Banach algebras

Matej Brešar, Yuri V. Turovskii (2009)

Studia Mathematica

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We consider a continuous derivation D on a Banach algebra 𝓐 such that p(D) is a compact operator for some polynomial p. It is shown that either 𝓐 has a nonzero finite-dimensional ideal not contained in the radical rad(𝓐) of 𝓐 or there exists another polynomial p̃ such that p̃(D) maps 𝓐 into rad(𝓐). A special case where Dⁿ is compact is discussed in greater detail.

Properties of the spectral radius in Banach algebras

Jaroslav Zemánek (1982)

Banach Center Publications

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Range inclusion results for derivations on noncommutative Banach algebras

Volker Runde (1993)

Studia Mathematica

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Let A be a Banach algebra, and let D : A → A be a (possibly unbounded) derivation. We are interested in two problems concerning the range of D: 1. When does D map into the (Jacobson) radical of A? 2. If [a,Da] = 0 for some a ∈ A, is Da necessarily quasinilpotent? We prove that derivations satisfying certain polynomial identities map into the radical. As an application, we show that if [a,[a,[a,Da]]] lies in the prime radical of A for all a ∈ A, then D maps into the radical. This generalizes...

An analytic approach to the spectral characterization of the radical

Graham R. Allan (2008)

Studia Mathematica

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We give a simple complex-variable proof of an old result of Zemánek and Le Page on the radical of a Banach algebra. Incidentally, the argument also proves a recent result of Harris and Kadison.

Time-dependent perturbation theory for abstract evolution equations of second order

Yuhua Lin (1998)

Studia Mathematica

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A condition on a family $B (t) : t \in [0, T]$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation $u_{t t} = u_{x x} + b (t, x) u_{x} (t, x) + c (t, x) u (t, x) + f (t, x) f o r (t, x) \in [0, T] \times [0, 1], u (t, 0) = u (t, 1) = 0 f o r t \in [0, T], u (0, x) = u_{0} (x), u_{t} (0, x) = v_{0} (x) f o r x \in [0, 1]$ in the space of continuous functions on [0,1].