Banach Algebra Elements Research Articles

On Ideals of Generalised Invertible Elements in Banach Algebras

We characterise ideals consisting of generalised invertible elements in a semisimple Banach algebra.

SOME RESULTS FOR QUADRATIC ELEMENTS OF A BANACH ALGEBRA

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Several properties of some quadratic elements of a unitial Banach algebra are studied. Deddens subspaces are also introduced and discussed.

Convergence Properties of Positive Elements in Banach Algebras

We recall the definition and properties of an algebra cone in an ordered Banach algebra and continue to develop spectral theory for the positive elements. If ($a_{n}$) is a sequence of positive elements converging to a, then an interesting question is that of which properties of the spectral radius r(a) of a are 'inherited' by $r(a_{n})$. We show that under suitable circumstances if r(a) is a Riesz point of the spectrum σ(a) of a (relative to some inessential ideal), then $r(a_{n})\rightarrow r(a)$ and, for all n big enough, $r(a_{n})$ is a Riesz point of $\sigma (a_{n})$. If the Laurent series of the corresponding resolvents are then investigated, some conclusions can be drawn regarding the convergence of the spectral idempotents, as well as the positive eigenvectors associated with $a_{n}$. Some of these results are applicable to certain types of operators.

Convergence Properties of Positive Elements in Banach Algebras

Convergence Properties of Positive Elements in Banach Algebras

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C *-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.

ALL PRIMES HAVE CLOSED RANGE

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.

The perturbation theory for the Drazin inverse and its applications II

AbstractWe study the perturbation of the generalized Drazin inverse for the elements of Banach algebras and bounded linear operators on Banach space. This work, among other things, extends the results obtained by the second author and Guorong Wang on the Drazin inverse for matrices.

Binomial sums, moments and invariant subspaces

The main result of this paper is that if a sequence of complex numbers (an)n≥0 satisfies\(\sum\limits_{\mathop {k = 0}\limits_{k even} }^n {\left({_k^n } \right)a_k = O(n^r)} \)and % MathType!MTEF!2!1!+-\(\sum\limits_{\mathop {k = 0}\limits_{k odd} }^n {\left({_k^n } \right)a_k = O(n^r )} as n \to \infty,\) for some integerr≥0, thenan=0 for alln>r. As an application, we deduce a localized form of a theorem of Allan about nilpotent elements in Banach algebras, and this in turn leads to an invariant-subspace theorem. As a further application, we prove a variant of Carleman's theorem on the unique determination of probability distributions by their moments. The paper concludes with a quantitative form of the main result.

Perturbation theorems for Hermitian elements in Banach algebras

Two well-known theorems for Hermitian elements in C*-algebras are extended to Banach algebras. The first concerns the solution of the equation ax - xb = y, and the second gives sharp bounds for the distance between spectra of a and b when a, b are Hermiti

Continuity of Lie mappings of the skew elements of Banach algebras with involution

Let A A and B B be centrally closed prime complex Banach algebras with linear involution. If A A is semisimple, then any Lie derivation of the skew elements of A A is continuous and any Lie isomorphism from the skew elements of B B onto the skew elements of A A is continuous.

Numerical ranges of elements of involutive Banach algebras and commutativity

Let $ {\cal A} $ be a unital involutive Banach algebra. Bonsall and Duncan defined the numerical range for each element x in $ {\cal A} $ by $ V(x) = \{f(x):f \in {\cal A'}, f(e) = 1 =|| f || \} $ , where e is the unit. We introduce another numerical range by $ W (x) = \{ f(x) : f \in {\cal A'}, f \geq 0, f(e) = 1 \} $ , and we call $ w(x) = {\rm sup} \{|z|: z \in W (x) \} $ the numerical radius of x. We give a few conditions for $ {\cal A} $ to be a C *-algebra, and we see that some mapping theorems for numerical ranges of elements of $ {\cal A} $ hold. We show that if w (x) ≤ 1 implies $ w (\varphi (x) ) \leq 1 $ for every Mobius transform $ \varphi $ of the unit disk, then $ {\cal A} $ is commutative.

On the joint spectral radius

We prove the `p-spectral radius formula for n-tuples of commuting Banach algebra elements. This generalizes results of some earlier papers. Let A be a Banach algebra with the unit element denoted by 1. Let a = (a1, . . . , an) be an n-tuple of elements of A. Denote by σ(a) the Harte spectrum of a, i.e. λ = (λ1, . . . , λn) 6∈ σ(a) if and only if there exist u1, . . . , un, v1, . . . , vn ∈ A such that n ∑ j=1 (aj − λj)uj = n ∑ j=1 vj(aj − λj) = 1. Let 1 ≤ p ≤ ∞. The (geometric) spectral radius of a is defined by rp(a) = max{‖λ‖p : λ ∈ σ(a)}, where ‖λ‖p = { max1≤j≤n |λj | (p =∞), ( ∑n j=1 |λj |) (1 ≤ p <∞); see [10], cf. also [4]. If σ(a) is empty we put formally rp(a) = −∞. Clearly, rp(a) depends on p. On the other hand, instead of the Harte spectrum we can take any other reasonable spectrum (e.g. the left, right, approximate point, defect, Taylor etc.) without changing the value of rp(a); see [4], [9]. For a single Banach algebra element the just defined spectral radius rp(a) does not depend on p and coincides with the ordinary spectral radius r(a1) = max{|λ1| : λ1 ∈ σ(a1)}. By the well-known spectral radius formula 1991 Mathematics Subject Classification: Primary 46H05, 46J05.

Spectral sets

The paper studies spectral sets of elements of Banach algebras as the zeros of holomorphic functions and describes them in terms of existence of idempotents. A new decomposition theorem characterizing spectral sets is obtained for bounded linear operators

On the differences of the consecutive powers of Banach algebra elements

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence {x(x − 1)}n∈N for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of {x}n∈N and { 1 n ∑n−1 k=0 x k } n∈N.

Functions from the Schoenberg classT act in the cone of dissipative elements of a Banach algebra. IIact in the cone of dissipative elements of a Banach algebra. II

The functional calculus of several commuting dissipative elements of a complex Banach algebra with identity, first introduced in the preceding work by the author, is developed. Uniqueness, continuity, and stability theorems, composite function theorems, and a formula for the resolvent are established. Applications to the theory of sectorial operators in Hilbert space are given.

Schurian Algebras and Spectral Additivity

A class of algebras is introduced that includes the unital Banach algebras over the complex numbers. Commutator results are proved for such algebras and used to establish spectral properties of certain elements of Banach algebras.

Spectral characterizations of central elements in Banach algebras

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).

Isolated spectral points

The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that 0 0 is isolated in the spectrum of a bounded linear operator T T if the (not necessarily closed) space M = { x : lim n ‖ T n x ‖ 1 / n = 0 } M=\{x: \lim _{n}\|T^nx\|^{1/n}=0\} is nonzero and complemented by a closed subspace N N satisfying T N ⊂ N ⊂ T X TN\subset N\subset TX .

Affine mappings of invertible operators

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

Formulas for the Joint Spectral Radius of Non-Commuting Banach Algebra Elements

Formulas for the Joint Spectral Radius of Non-Commuting Banach Algebra Elements