Topologically Invertible Elements and Topological Spectrum

Mati Abel; Wiesław Żelazko

Displaying similar documents to “Topologically Invertible Elements and Topological Spectrum”

Spectrum of L

W. Marek, K. Rasmussen

Similarity:

CONTENTS0. Motivation, results to be used in the sequel ................51. Slicing $L_{α}$ ’s ..........................................................102. Hereditarily countable, definable elements ................133. Spectrum of L.............................................................154. The width of elements of spectrum ............................195. Non-uniform strong definability ..................................266. Solution to a problem of Wilmers................................327....

Fermi Golden Rule, Feshbach Method and embedded point spectrum

Jan Dereziński (1998-1999)

Séminaire Équations aux dérivées partielles

Similarity:

A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak $\overset{ˇ}{s}$ ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

Similarity:

Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , $S p (A, B) = S p (A) \times S p (B)$ , where Sp denotes the joint Taylor spectrum.

Continuous characters and joint topological spectrum

Wiesław Żelazko (2007)

Control and Cybernetics

Similarity:

Ascent spectrum and essential ascent spectrum

O. Bel Hadj Fredj, M. Burgos, M. Oudghiri (2008)

Studia Mathematica

Similarity:

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if $σ_{a s c}^{e} (T + F) = σ_{a s c}^{e} (T)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued...

The norm spectrum in certain classes of commutative Banach algebras

H. S. Mustafayev (2011)

Colloquium Mathematicae

Similarity:

Let A be a commutative Banach algebra and let $Σ_{A}$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $σ (f) = \bar{f \cdot a : a \in A} \cap Σ_{A}$ , where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

A note on the singular spectrum.

L. Lindeboom (Groenewald), H. Raubenheimer (1998)

Extracta Mathematicae

Similarity:

We compare the singular spectrum of a Banach algebra element with the usual spectrum and exponential spectrum.

On certain products of Banach algebras with applications to harmonic analysis

Mehdi Sangani Monfared (2007)

Studia Mathematica

Similarity:

Given Banach algebras A and B with spectrum σ(B) ≠ ∅, and given θ ∈ σ(B), we define a product $A \times_{θ} B$ , which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in ₀(X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral...

Generalized spectral perturbation and the boundary spectrum

Sonja Mouton (2021)

Czechoslovak Mathematical Journal

Similarity:

By considering arbitrary mappings $ω$ from a Banach algebra $A$ into the set of all nonempty, compact subsets of the complex plane such that for all $a \in A$ , the set $ω (a)$ lies between the boundary and connected hull of the exponential spectrum of $a$ , we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

Single valued extension property and generalized Weyl’s theorem

M. Berkani, N. Castro, S. V. Djordjević (2006)

Mathematica Bohemica

Similarity:

Let $T$ be an operator acting on a Banach space $X$ , let $σ (T)$ and $σ_{B W} (T)$ be respectively the spectrum and the B-Weyl spectrum of $T$ . We say that $T$ satisfies the generalized Weyl’s theorem if $σ_{B W} (T) = σ (T) ∖ E (T)$ , where $E (T)$ is the set of all isolated eigenvalues of $T$ . The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$ , extending an earlier result of J. K. Finch...

Local spectrum and local spectral radius of an operator at a fixed vector

Janko Bračič, Vladimír Müller (2009)

Studia Mathematica

Similarity:

Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with $σ_{T} (e) = σ_{δ} (T)$ , respectively $r_{T} (e) = r (T)$ , is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Three spectral notions for representations of commutative Banach algebras

Yngve Domar, Lars-Ake Lindahl (1975)

Annales de l'institut Fourier

Similarity:

Let $T$ be a bounded representation of a commutative Banach algebra $B$ . The following spectral sets are studied. $Λ_{1} (T)$ : the Gelfand space of the quotient algebra $B / Ker T$ . $Λ_{2} (T)$ : the Gelfand space of the operator algebra $Im T$ . $Λ_{3} (T)$ : those characters $φ$ of $B$ for which the inequalities $∥ T_{b} x - \hat{b} (φ) x ∥ < ϵ ∥ x ∥$ , $b \in F$ , have a common solution $x \neq 0$ , for any $ϵ > 0$ and any finite subset $F$ of $B$ . A theorem of Beurling on the spectrum of $L^{\infty}$ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of...